Integrand size = 16, antiderivative size = 225 \[ \int (c+d x)^4 \cosh ^3(a+b x) \, dx=-\frac {160 d^3 (c+d x) \cosh (a+b x)}{9 b^4}-\frac {8 d (c+d x)^3 \cosh (a+b x)}{3 b^2}-\frac {8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {488 d^4 \sinh (a+b x)}{27 b^5}+\frac {80 d^2 (c+d x)^2 \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac {4 d^2 (c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^4 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {8 d^4 \sinh ^3(a+b x)}{81 b^5} \] Output:
-160/9*d^3*(d*x+c)*cosh(b*x+a)/b^4-8/3*d*(d*x+c)^3*cosh(b*x+a)/b^2-8/27*d^ 3*(d*x+c)*cosh(b*x+a)^3/b^4-4/9*d*(d*x+c)^3*cosh(b*x+a)^3/b^2+488/27*d^4*s inh(b*x+a)/b^5+80/9*d^2*(d*x+c)^2*sinh(b*x+a)/b^3+2/3*(d*x+c)^4*sinh(b*x+a )/b+4/9*d^2*(d*x+c)^2*cosh(b*x+a)^2*sinh(b*x+a)/b^3+1/3*(d*x+c)^4*cosh(b*x +a)^2*sinh(b*x+a)/b+8/81*d^4*sinh(b*x+a)^3/b^5
Time = 0.73 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.70 \[ \int (c+d x)^4 \cosh ^3(a+b x) \, dx=\frac {-972 b d (c+d x) \left (6 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)-12 b d (c+d x) \left (2 d^2+3 b^2 (c+d x)^2\right ) \cosh (3 (a+b x))+2 \left (2920 d^4+1476 b^2 d^2 (c+d x)^2+135 b^4 (c+d x)^4+\left (8 d^4+36 b^2 d^2 (c+d x)^2+27 b^4 (c+d x)^4\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{324 b^5} \] Input:
Integrate[(c + d*x)^4*Cosh[a + b*x]^3,x]
Output:
(-972*b*d*(c + d*x)*(6*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] - 12*b*d*(c + d*x)*(2*d^2 + 3*b^2*(c + d*x)^2)*Cosh[3*(a + b*x)] + 2*(2920*d^4 + 1476*b^ 2*d^2*(c + d*x)^2 + 135*b^4*(c + d*x)^4 + (8*d^4 + 36*b^2*d^2*(c + d*x)^2 + 27*b^4*(c + d*x)^4)*Cosh[2*(a + b*x)])*Sinh[a + b*x])/(324*b^5)
Result contains complex when optimal does not.
Time = 1.86 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.47, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.688, Rules used = {3042, 3792, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 3792, 3042, 3113, 2009, 3777, 26, 3042, 26, 3777, 3042, 3117}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (c+d x)^4 \cosh ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^4 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \cosh ^3(a+b x)dx}{3 b^2}+\frac {2}{3} \int (c+d x)^4 \cosh (a+b x)dx-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \int (c+d x)^4 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}-\frac {4 i d \int -i (c+d x)^3 \sinh (a+b x)dx}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}-\frac {4 d \int (c+d x)^3 \sinh (a+b x)dx}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}-\frac {4 d \int -i (c+d x)^3 \sin (i a+i b x)dx}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \int (c+d x)^3 \sin (i a+i b x)dx}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \int (c+d x)^2 \cosh (a+b x)dx}{b}\right )}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 i d \int -i (c+d x) \sinh (a+b x)dx}{b}\right )}{b}\right )}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 d \int (c+d x) \sinh (a+b x)dx}{b}\right )}{b}\right )}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 d \int -i (c+d x) \sin (i a+i b x)dx}{b}\right )}{b}\right )}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \int (c+d x) \sin (i a+i b x)dx}{b}\right )}{b}\right )}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \cosh (a+b x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )}{b}\right )}{b}\right )-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {4 d^2 \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2 d^2 \int \cosh ^3(a+b x)dx}{9 b^2}+\frac {2}{3} \int (c+d x)^2 \cosh (a+b x)dx-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2 d^2 \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^3dx}{9 b^2}+\frac {2}{3} \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2 i d^2 \int \left (\sinh ^2(a+b x)+1\right )d(-i \sinh (a+b x))}{9 b^3}+\frac {2}{3} \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2}{3} \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx+\frac {2 i d^2 \left (-\frac {1}{3} i \sinh ^3(a+b x)-i \sinh (a+b x)\right )}{9 b^3}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 i d \int -i (c+d x) \sinh (a+b x)dx}{b}\right )+\frac {2 i d^2 \left (-\frac {1}{3} i \sinh ^3(a+b x)-i \sinh (a+b x)\right )}{9 b^3}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 d \int (c+d x) \sinh (a+b x)dx}{b}\right )+\frac {2 i d^2 \left (-\frac {1}{3} i \sinh ^3(a+b x)-i \sinh (a+b x)\right )}{9 b^3}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 d \int -i (c+d x) \sin (i a+i b x)dx}{b}\right )+\frac {2 i d^2 \left (-\frac {1}{3} i \sinh ^3(a+b x)-i \sinh (a+b x)\right )}{9 b^3}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \int (c+d x) \sin (i a+i b x)dx}{b}\right )+\frac {2 i d^2 \left (-\frac {1}{3} i \sinh ^3(a+b x)-i \sinh (a+b x)\right )}{9 b^3}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \cosh (a+b x)dx}{b}\right )}{b}\right )+\frac {2 i d^2 \left (-\frac {1}{3} i \sinh ^3(a+b x)-i \sinh (a+b x)\right )}{9 b^3}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )+\frac {2 i d^2 \left (-\frac {1}{3} i \sinh ^3(a+b x)-i \sinh (a+b x)\right )}{9 b^3}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\right )+\frac {4 d^2 \left (\frac {2 i d^2 \left (-\frac {1}{3} i \sinh ^3(a+b x)-i \sinh (a+b x)\right )}{9 b^3}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b}\) |
Input:
Int[(c + d*x)^4*Cosh[a + b*x]^3,x]
Output:
(-4*d*(c + d*x)^3*Cosh[a + b*x]^3)/(9*b^2) + ((c + d*x)^4*Cosh[a + b*x]^2* Sinh[a + b*x])/(3*b) + (4*d^2*((-2*d*(c + d*x)*Cosh[a + b*x]^3)/(9*b^2) + ((c + d*x)^2*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b) + (((2*I)/9)*d^2*((-I)*S inh[a + b*x] - (I/3)*Sinh[a + b*x]^3))/b^3 + (2*(((c + d*x)^2*Sinh[a + b*x ])/b + ((2*I)*d*((I*(c + d*x)*Cosh[a + b*x])/b - (I*d*Sinh[a + b*x])/b^2)) /b))/3))/(3*b^2) + (2*(((c + d*x)^4*Sinh[a + b*x])/b + ((4*I)*d*((I*(c + d *x)^3*Cosh[a + b*x])/b - ((3*I)*d*(((c + d*x)^2*Sinh[a + b*x])/b + ((2*I)* d*((I*(c + d*x)*Cosh[a + b*x])/b - (I*d*Sinh[a + b*x])/b^2))/b))/b))/b))/3
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Time = 2.06 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.77
| method | result | size |
| parallelrisch | \(\frac {\left (27 \left (d x +c \right )^{4} b^{4}+36 d^{2} \left (d x +c \right )^{2} b^{2}+8 d^{4}\right ) \sinh \left (3 b x +3 a \right )-36 \left (d x +c \right ) b \left (\left (d x +c \right )^{2} b^{2}+\frac {2 d^{2}}{3}\right ) d \cosh \left (3 b x +3 a \right )+243 \left (\left (d x +c \right )^{4} b^{4}+12 d^{2} \left (d x +c \right )^{2} b^{2}+24 d^{4}\right ) \sinh \left (b x +a \right )-972 \left (\left (\left (d x +c \right )^{2} b^{2}+6 d^{2}\right ) \left (d x +c \right ) \cosh \left (b x +a \right )+\frac {28 b^{2} c^{3}}{27}+\frac {488 d^{2} c}{81}\right ) b d}{324 b^{5}}\) | \(173\) |
| risch | \(\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}-36 b^{3} d^{4} x^{3}+108 b^{4} c^{3} d x -108 b^{3} c \,d^{3} x^{2}+27 b^{4} c^{4}-108 b^{3} c^{2} d^{2} x +36 b^{2} d^{4} x^{2}-36 b^{3} c^{3} d +72 b^{2} c \,d^{3} x +36 b^{2} c^{2} d^{2}-24 b \,d^{4} x -24 b c \,d^{3}+8 d^{4}\right ) {\mathrm e}^{3 b x +3 a}}{648 b^{5}}+\frac {3 \left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}-4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x -12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}-12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}-4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}-24 b \,d^{4} x -24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{b x +a}}{8 b^{5}}-\frac {3 \left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x +12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}+12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}+4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}+24 b \,d^{4} x +24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{-b x -a}}{8 b^{5}}-\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+36 b^{3} d^{4} x^{3}+108 b^{4} c^{3} d x +108 b^{3} c \,d^{3} x^{2}+27 b^{4} c^{4}+108 b^{3} c^{2} d^{2} x +36 b^{2} d^{4} x^{2}+36 b^{3} c^{3} d +72 b^{2} c \,d^{3} x +36 b^{2} c^{2} d^{2}+24 b \,d^{4} x +24 b c \,d^{3}+8 d^{4}\right ) {\mathrm e}^{-3 b x -3 a}}{648 b^{5}}\) | \(655\) |
| orering | \(-\frac {16 d \left (135 b^{6} d^{6} x^{6}+810 b^{6} c \,d^{5} x^{5}+2025 b^{6} c^{2} d^{4} x^{4}+2700 b^{6} c^{3} d^{3} x^{3}+2025 b^{6} c^{4} d^{2} x^{2}+891 b^{4} d^{6} x^{4}+810 b^{6} c^{5} d x +3564 b^{4} c \,d^{5} x^{3}+135 b^{6} c^{6}+5346 b^{4} c^{2} d^{4} x^{2}+3564 b^{4} c^{3} d^{3} x +891 b^{4} c^{4} d^{2}-1960 b^{2} d^{6} x^{2}-3920 b^{2} c \,d^{5} x -1960 b^{2} c^{2} d^{4}-5460 d^{6}\right ) \cosh \left (b x +a \right )^{3}}{243 b^{8} \left (d x +c \right )^{3}}+\frac {2 \left (135 b^{6} d^{6} x^{6}+810 b^{6} c \,d^{5} x^{5}+2025 b^{6} c^{2} d^{4} x^{4}+2700 b^{6} c^{3} d^{3} x^{3}+2025 b^{6} c^{4} d^{2} x^{2}+396 b^{4} d^{6} x^{4}+810 b^{6} c^{5} d x +1584 b^{4} c \,d^{5} x^{3}+135 b^{6} c^{6}+2376 b^{4} c^{2} d^{4} x^{2}+1584 b^{4} c^{3} d^{3} x +396 b^{4} c^{4} d^{2}-10400 b^{2} d^{6} x^{2}-20800 b^{2} c \,d^{5} x -10400 b^{2} c^{2} d^{4}-21840 d^{6}\right ) \left (4 \left (d x +c \right )^{3} \cosh \left (b x +a \right )^{3} d +3 \left (d x +c \right )^{4} \cosh \left (b x +a \right )^{2} b \sinh \left (b x +a \right )\right )}{243 b^{8} \left (d x +c \right )^{6}}+\frac {16 d \left (9 d^{4} x^{4} b^{4}+36 b^{4} c \,d^{3} x^{3}+54 b^{4} c^{2} d^{2} x^{2}+36 b^{4} c^{3} d x +9 b^{4} c^{4}+105 b^{2} d^{4} x^{2}+210 b^{2} c \,d^{3} x +105 b^{2} c^{2} d^{2}+182 d^{4}\right ) \left (12 \left (d x +c \right )^{2} \cosh \left (b x +a \right )^{3} d^{2}+24 \left (d x +c \right )^{3} \cosh \left (b x +a \right )^{2} d b \sinh \left (b x +a \right )+6 \left (d x +c \right )^{4} \cosh \left (b x +a \right ) b^{2} \sinh \left (b x +a \right )^{2}+3 \left (d x +c \right )^{4} \cosh \left (b x +a \right )^{3} b^{2}\right )}{81 b^{8} \left (d x +c \right )^{5}}-\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+108 b^{4} c^{3} d x +27 b^{4} c^{4}+360 b^{2} d^{4} x^{2}+720 b^{2} c \,d^{3} x +360 b^{2} c^{2} d^{2}+728 d^{4}\right ) \left (24 \left (d x +c \right ) \cosh \left (b x +a \right )^{3} d^{3}+108 \left (d x +c \right )^{2} \cosh \left (b x +a \right )^{2} d^{2} b \sinh \left (b x +a \right )+72 \left (d x +c \right )^{3} \cosh \left (b x +a \right ) d \,b^{2} \sinh \left (b x +a \right )^{2}+36 \left (d x +c \right )^{3} \cosh \left (b x +a \right )^{3} d \,b^{2}+6 \left (d x +c \right )^{4} b^{3} \sinh \left (b x +a \right )^{3}+21 \left (d x +c \right )^{4} \cosh \left (b x +a \right )^{2} b^{3} \sinh \left (b x +a \right )\right )}{243 b^{8} \left (d x +c \right )^{4}}\) | \(883\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1139\) |
| default | \(\text {Expression too large to display}\) | \(1139\) |
Input:
int((d*x+c)^4*cosh(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/324*((27*(d*x+c)^4*b^4+36*d^2*(d*x+c)^2*b^2+8*d^4)*sinh(3*b*x+3*a)-36*(d *x+c)*b*((d*x+c)^2*b^2+2/3*d^2)*d*cosh(3*b*x+3*a)+243*((d*x+c)^4*b^4+12*d^ 2*(d*x+c)^2*b^2+24*d^4)*sinh(b*x+a)-972*(((d*x+c)^2*b^2+6*d^2)*(d*x+c)*cos h(b*x+a)+28/27*b^2*c^3+488/81*d^2*c)*b*d)/b^5
Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (205) = 410\).
Time = 0.09 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.35 \[ \int (c+d x)^4 \cosh ^3(a+b x) \, dx=-\frac {12 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right )^{3} + 36 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \sinh \left (b x + a\right )^{3} + 972 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d + 6 \, b c d^{3} + 3 \, {\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right ) - 3 \, {\left (81 \, b^{4} d^{4} x^{4} + 324 \, b^{4} c d^{3} x^{3} + 81 \, b^{4} c^{4} + 972 \, b^{2} c^{2} d^{2} + 1944 \, d^{4} + 486 \, {\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{2} + 324 \, {\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{324 \, b^{5}} \] Input:
integrate((d*x+c)^4*cosh(b*x+a)^3,x, algorithm="fricas")
Output:
-1/324*(12*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d + 2*b*c*d^3 + (9 *b^3*c^2*d^2 + 2*b*d^4)*x)*cosh(b*x + a)^3 + 36*(3*b^3*d^4*x^3 + 9*b^3*c*d ^3*x^2 + 3*b^3*c^3*d + 2*b*c*d^3 + (9*b^3*c^2*d^2 + 2*b*d^4)*x)*cosh(b*x + a)*sinh(b*x + a)^2 - (27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 + 3 6*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*c^2*d^2 + 2*b^2*d^4)*x^2 + 36*(3*b^4*c^3 *d + 2*b^2*c*d^3)*x)*sinh(b*x + a)^3 + 972*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + b^3*c^3*d + 6*b*c*d^3 + 3*(b^3*c^2*d^2 + 2*b*d^4)*x)*cosh(b*x + a) - 3*( 81*b^4*d^4*x^4 + 324*b^4*c*d^3*x^3 + 81*b^4*c^4 + 972*b^2*c^2*d^2 + 1944*d ^4 + 486*(b^4*c^2*d^2 + 2*b^2*d^4)*x^2 + (27*b^4*d^4*x^4 + 108*b^4*c*d^3*x ^3 + 27*b^4*c^4 + 36*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*c^2*d^2 + 2*b^2*d^4)* x^2 + 36*(3*b^4*c^3*d + 2*b^2*c*d^3)*x)*cosh(b*x + a)^2 + 324*(b^4*c^3*d + 6*b^2*c*d^3)*x)*sinh(b*x + a))/b^5
Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (226) = 452\).
Time = 0.65 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.43 \[ \int (c+d x)^4 \cosh ^3(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)**4*cosh(b*x+a)**3,x)
Output:
Piecewise((-2*c**4*sinh(a + b*x)**3/(3*b) + c**4*sinh(a + b*x)*cosh(a + b* x)**2/b - 8*c**3*d*x*sinh(a + b*x)**3/(3*b) + 4*c**3*d*x*sinh(a + b*x)*cos h(a + b*x)**2/b - 4*c**2*d**2*x**2*sinh(a + b*x)**3/b + 6*c**2*d**2*x**2*s inh(a + b*x)*cosh(a + b*x)**2/b - 8*c*d**3*x**3*sinh(a + b*x)**3/(3*b) + 4 *c*d**3*x**3*sinh(a + b*x)*cosh(a + b*x)**2/b - 2*d**4*x**4*sinh(a + b*x)* *3/(3*b) + d**4*x**4*sinh(a + b*x)*cosh(a + b*x)**2/b + 8*c**3*d*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**2) - 28*c**3*d*cosh(a + b*x)**3/(9*b**2) + 8*c **2*d**2*x*sinh(a + b*x)**2*cosh(a + b*x)/b**2 - 28*c**2*d**2*x*cosh(a + b *x)**3/(3*b**2) + 8*c*d**3*x**2*sinh(a + b*x)**2*cosh(a + b*x)/b**2 - 28*c *d**3*x**2*cosh(a + b*x)**3/(3*b**2) + 8*d**4*x**3*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**2) - 28*d**4*x**3*cosh(a + b*x)**3/(9*b**2) - 80*c**2*d**2*s inh(a + b*x)**3/(9*b**3) + 28*c**2*d**2*sinh(a + b*x)*cosh(a + b*x)**2/(3* b**3) - 160*c*d**3*x*sinh(a + b*x)**3/(9*b**3) + 56*c*d**3*x*sinh(a + b*x) *cosh(a + b*x)**2/(3*b**3) - 80*d**4*x**2*sinh(a + b*x)**3/(9*b**3) + 28*d **4*x**2*sinh(a + b*x)*cosh(a + b*x)**2/(3*b**3) + 160*c*d**3*sinh(a + b*x )**2*cosh(a + b*x)/(9*b**4) - 488*c*d**3*cosh(a + b*x)**3/(27*b**4) + 160* d**4*x*sinh(a + b*x)**2*cosh(a + b*x)/(9*b**4) - 488*d**4*x*cosh(a + b*x)* *3/(27*b**4) - 1456*d**4*sinh(a + b*x)**3/(81*b**5) + 488*d**4*sinh(a + b* x)*cosh(a + b*x)**2/(27*b**5), Ne(b, 0)), ((c**4*x + 2*c**3*d*x**2 + 2*c** 2*d**2*x**3 + c*d**3*x**4 + d**4*x**5/5)*cosh(a)**3, True))
Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (205) = 410\).
Time = 0.09 (sec) , antiderivative size = 644, normalized size of antiderivative = 2.86 \[ \int (c+d x)^4 \cosh ^3(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^4*cosh(b*x+a)^3,x, algorithm="maxima")
Output:
1/18*c^3*d*((3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 + 27*(b*x*e^a - e^a)*e ^(b*x)/b^2 - 27*(b*x + 1)*e^(-b*x - a)/b^2 - (3*b*x + 1)*e^(-3*b*x - 3*a)/ b^2) + 1/24*c^4*(e^(3*b*x + 3*a)/b + 9*e^(b*x + a)/b - 9*e^(-b*x - a)/b - e^(-3*b*x - 3*a)/b) + 1/36*c^2*d^2*((9*b^2*x^2*e^(3*a) - 6*b*x*e^(3*a) + 2 *e^(3*a))*e^(3*b*x)/b^3 + 81*(b^2*x^2*e^a - 2*b*x*e^a + 2*e^a)*e^(b*x)/b^3 - 81*(b^2*x^2 + 2*b*x + 2)*e^(-b*x - a)/b^3 - (9*b^2*x^2 + 6*b*x + 2)*e^( -3*b*x - 3*a)/b^3) + 1/54*c*d^3*((9*b^3*x^3*e^(3*a) - 9*b^2*x^2*e^(3*a) + 6*b*x*e^(3*a) - 2*e^(3*a))*e^(3*b*x)/b^4 + 81*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*e^(b*x)/b^4 - 81*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e ^(-b*x - a)/b^4 - (9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a)/b^4 ) + 1/648*d^4*((27*b^4*x^4*e^(3*a) - 36*b^3*x^3*e^(3*a) + 36*b^2*x^2*e^(3* a) - 24*b*x*e^(3*a) + 8*e^(3*a))*e^(3*b*x)/b^5 + 243*(b^4*x^4*e^a - 4*b^3* x^3*e^a + 12*b^2*x^2*e^a - 24*b*x*e^a + 24*e^a)*e^(b*x)/b^5 - 243*(b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*e^(-b*x - a)/b^5 - (27*b^4*x^4 + 36*b^3*x^3 + 36*b^2*x^2 + 24*b*x + 8)*e^(-3*b*x - 3*a)/b^5)
Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (205) = 410\).
Time = 0.13 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.91 \[ \int (c+d x)^4 \cosh ^3(a+b x) \, dx=\frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} - 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x - 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} - 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} - 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{648 \, b^{5}} + \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{8 \, b^{5}} - \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{8 \, b^{5}} - \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x + 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} + 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} + 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{648 \, b^{5}} \] Input:
integrate((d*x+c)^4*cosh(b*x+a)^3,x, algorithm="giac")
Output:
1/648*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 - 36*b^3*d ^4*x^3 + 108*b^4*c^3*d*x - 108*b^3*c*d^3*x^2 + 27*b^4*c^4 - 108*b^3*c^2*d^ 2*x + 36*b^2*d^4*x^2 - 36*b^3*c^3*d + 72*b^2*c*d^3*x + 36*b^2*c^2*d^2 - 24 *b*d^4*x - 24*b*c*d^3 + 8*d^4)*e^(3*b*x + 3*a)/b^5 + 3/8*(b^4*d^4*x^4 + 4* b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 - 4*b^3*d^4*x^3 + 4*b^4*c^3*d*x - 12*b^3 *c*d^3*x^2 + b^4*c^4 - 12*b^3*c^2*d^2*x + 12*b^2*d^4*x^2 - 4*b^3*c^3*d + 2 4*b^2*c*d^3*x + 12*b^2*c^2*d^2 - 24*b*d^4*x - 24*b*c*d^3 + 24*d^4)*e^(b*x + a)/b^5 - 3/8*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^3* d^4*x^3 + 4*b^4*c^3*d*x + 12*b^3*c*d^3*x^2 + b^4*c^4 + 12*b^3*c^2*d^2*x + 12*b^2*d^4*x^2 + 4*b^3*c^3*d + 24*b^2*c*d^3*x + 12*b^2*c^2*d^2 + 24*b*d^4* x + 24*b*c*d^3 + 24*d^4)*e^(-b*x - a)/b^5 - 1/648*(27*b^4*d^4*x^4 + 108*b^ 4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 + 36*b^3*d^4*x^3 + 108*b^4*c^3*d*x + 108 *b^3*c*d^3*x^2 + 27*b^4*c^4 + 108*b^3*c^2*d^2*x + 36*b^2*d^4*x^2 + 36*b^3* c^3*d + 72*b^2*c*d^3*x + 36*b^2*c^2*d^2 + 24*b*d^4*x + 24*b*c*d^3 + 8*d^4) *e^(-3*b*x - 3*a)/b^5
Time = 2.44 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.36 \[ \int (c+d x)^4 \cosh ^3(a+b x) \, dx=\frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (27\,b^4\,c^4+252\,b^2\,c^2\,d^2+488\,d^4\right )}{27\,b^5}-\frac {2\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (27\,b^4\,c^4+360\,b^2\,c^2\,d^2+728\,d^4\right )}{81\,b^5}-\frac {4\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (21\,b^2\,c^3\,d+122\,c\,d^3\right )}{27\,b^4}+\frac {8\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (3\,b^2\,c^3\,d+20\,c\,d^3\right )}{9\,b^4}-\frac {28\,d^4\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{9\,b^2}-\frac {4\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (63\,b^2\,c^2\,d^2+122\,d^4\right )}{27\,b^4}-\frac {2\,d^4\,x^4\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b}-\frac {8\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (3\,b^2\,c^3\,d+20\,c\,d^3\right )}{9\,b^3}-\frac {4\,x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (9\,b^2\,c^2\,d^2+20\,d^4\right )}{9\,b^3}+\frac {2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (9\,b^2\,c^2\,d^2+14\,d^4\right )}{3\,b^3}-\frac {28\,c\,d^3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b^2}+\frac {d^4\,x^4\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {8\,d^4\,x^3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{3\,b^2}-\frac {8\,c\,d^3\,x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b}+\frac {8\,x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^2\,c^2\,d^2+20\,d^4\right )}{9\,b^4}+\frac {4\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (3\,b^2\,c^3\,d+14\,c\,d^3\right )}{3\,b^3}+\frac {4\,c\,d^3\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {8\,c\,d^3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2} \] Input:
int(cosh(a + b*x)^3*(c + d*x)^4,x)
Output:
(cosh(a + b*x)^2*sinh(a + b*x)*(488*d^4 + 27*b^4*c^4 + 252*b^2*c^2*d^2))/( 27*b^5) - (2*sinh(a + b*x)^3*(728*d^4 + 27*b^4*c^4 + 360*b^2*c^2*d^2))/(81 *b^5) - (4*cosh(a + b*x)^3*(122*c*d^3 + 21*b^2*c^3*d))/(27*b^4) + (8*cosh( a + b*x)*sinh(a + b*x)^2*(20*c*d^3 + 3*b^2*c^3*d))/(9*b^4) - (28*d^4*x^3*c osh(a + b*x)^3)/(9*b^2) - (4*x*cosh(a + b*x)^3*(122*d^4 + 63*b^2*c^2*d^2)) /(27*b^4) - (2*d^4*x^4*sinh(a + b*x)^3)/(3*b) - (8*x*sinh(a + b*x)^3*(20*c *d^3 + 3*b^2*c^3*d))/(9*b^3) - (4*x^2*sinh(a + b*x)^3*(20*d^4 + 9*b^2*c^2* d^2))/(9*b^3) + (2*x^2*cosh(a + b*x)^2*sinh(a + b*x)*(14*d^4 + 9*b^2*c^2*d ^2))/(3*b^3) - (28*c*d^3*x^2*cosh(a + b*x)^3)/(3*b^2) + (d^4*x^4*cosh(a + b*x)^2*sinh(a + b*x))/b + (8*d^4*x^3*cosh(a + b*x)*sinh(a + b*x)^2)/(3*b^2 ) - (8*c*d^3*x^3*sinh(a + b*x)^3)/(3*b) + (8*x*cosh(a + b*x)*sinh(a + b*x) ^2*(20*d^4 + 9*b^2*c^2*d^2))/(9*b^4) + (4*x*cosh(a + b*x)^2*sinh(a + b*x)* (14*c*d^3 + 3*b^2*c^3*d))/(3*b^3) + (4*c*d^3*x^3*cosh(a + b*x)^2*sinh(a + b*x))/b + (8*c*d^3*x^2*cosh(a + b*x)*sinh(a + b*x)^2)/b^2
Time = 0.16 (sec) , antiderivative size = 1068, normalized size of antiderivative = 4.75 \[ \int (c+d x)^4 \cosh ^3(a+b x) \, dx=\frac {243 e^{4 b x +4 a} b^{4} d^{4} x^{4}-972 e^{4 b x +4 a} b^{3} c^{3} d -972 e^{4 b x +4 a} b^{3} d^{4} x^{3}+2916 e^{4 b x +4 a} b^{2} c^{2} d^{2}+5832 e^{4 b x +4 a} d^{4}-27 b^{4} c^{4}+972 e^{4 b x +4 a} b^{4} c^{3} d x +1458 e^{4 b x +4 a} b^{4} c^{2} d^{2} x^{2}+972 e^{4 b x +4 a} b^{4} c \,d^{3} x^{3}-2916 e^{4 b x +4 a} b^{3} c^{2} d^{2} x -2916 e^{4 b x +4 a} b^{3} c \,d^{3} x^{2}+5832 e^{4 b x +4 a} b^{2} c \,d^{3} x -5832 e^{2 b x +2 a} b^{2} c \,d^{3} x +8 e^{6 b x +6 a} d^{4}-5832 e^{2 b x +2 a} d^{4}+243 e^{4 b x +4 a} b^{4} c^{4}-27 b^{4} d^{4} x^{4}-36 b^{3} c^{3} d -36 b^{3} d^{4} x^{3}-36 b^{2} c^{2} d^{2}-36 b^{2} d^{4} x^{2}-24 b c \,d^{3}-24 b \,d^{4} x +27 e^{6 b x +6 a} b^{4} d^{4} x^{4}-36 e^{6 b x +6 a} b^{3} c^{3} d -36 e^{6 b x +6 a} b^{3} d^{4} x^{3}+36 e^{6 b x +6 a} b^{2} c^{2} d^{2}+36 e^{6 b x +6 a} b^{2} d^{4} x^{2}-24 e^{6 b x +6 a} b c \,d^{3}-24 e^{6 b x +6 a} b \,d^{4} x +2916 e^{4 b x +4 a} b^{2} d^{4} x^{2}-5832 e^{4 b x +4 a} b c \,d^{3}-5832 e^{4 b x +4 a} b \,d^{4} x -108 b^{4} c^{3} d x -162 b^{4} c^{2} d^{2} x^{2}-108 b^{4} c \,d^{3} x^{3}-108 b^{3} c^{2} d^{2} x -108 b^{3} c \,d^{3} x^{2}-72 b^{2} c \,d^{3} x -8 d^{4}-243 e^{2 b x +2 a} b^{4} d^{4} x^{4}-972 e^{2 b x +2 a} b^{3} c^{3} d -972 e^{2 b x +2 a} b^{3} d^{4} x^{3}-2916 e^{2 b x +2 a} b^{2} c^{2} d^{2}-2916 e^{2 b x +2 a} b^{2} d^{4} x^{2}-5832 e^{2 b x +2 a} b c \,d^{3}-5832 e^{2 b x +2 a} b \,d^{4} x +27 e^{6 b x +6 a} b^{4} c^{4}-243 e^{2 b x +2 a} b^{4} c^{4}+108 e^{6 b x +6 a} b^{4} c^{3} d x +162 e^{6 b x +6 a} b^{4} c^{2} d^{2} x^{2}+108 e^{6 b x +6 a} b^{4} c \,d^{3} x^{3}-108 e^{6 b x +6 a} b^{3} c^{2} d^{2} x -108 e^{6 b x +6 a} b^{3} c \,d^{3} x^{2}+72 e^{6 b x +6 a} b^{2} c \,d^{3} x -972 e^{2 b x +2 a} b^{4} c^{3} d x -1458 e^{2 b x +2 a} b^{4} c^{2} d^{2} x^{2}-972 e^{2 b x +2 a} b^{4} c \,d^{3} x^{3}-2916 e^{2 b x +2 a} b^{3} c^{2} d^{2} x -2916 e^{2 b x +2 a} b^{3} c \,d^{3} x^{2}}{648 e^{3 b x +3 a} b^{5}} \] Input:
int((d*x+c)^4*cosh(b*x+a)^3,x)
Output:
(27*e**(6*a + 6*b*x)*b**4*c**4 + 108*e**(6*a + 6*b*x)*b**4*c**3*d*x + 162* e**(6*a + 6*b*x)*b**4*c**2*d**2*x**2 + 108*e**(6*a + 6*b*x)*b**4*c*d**3*x* *3 + 27*e**(6*a + 6*b*x)*b**4*d**4*x**4 - 36*e**(6*a + 6*b*x)*b**3*c**3*d - 108*e**(6*a + 6*b*x)*b**3*c**2*d**2*x - 108*e**(6*a + 6*b*x)*b**3*c*d**3 *x**2 - 36*e**(6*a + 6*b*x)*b**3*d**4*x**3 + 36*e**(6*a + 6*b*x)*b**2*c**2 *d**2 + 72*e**(6*a + 6*b*x)*b**2*c*d**3*x + 36*e**(6*a + 6*b*x)*b**2*d**4* x**2 - 24*e**(6*a + 6*b*x)*b*c*d**3 - 24*e**(6*a + 6*b*x)*b*d**4*x + 8*e** (6*a + 6*b*x)*d**4 + 243*e**(4*a + 4*b*x)*b**4*c**4 + 972*e**(4*a + 4*b*x) *b**4*c**3*d*x + 1458*e**(4*a + 4*b*x)*b**4*c**2*d**2*x**2 + 972*e**(4*a + 4*b*x)*b**4*c*d**3*x**3 + 243*e**(4*a + 4*b*x)*b**4*d**4*x**4 - 972*e**(4 *a + 4*b*x)*b**3*c**3*d - 2916*e**(4*a + 4*b*x)*b**3*c**2*d**2*x - 2916*e* *(4*a + 4*b*x)*b**3*c*d**3*x**2 - 972*e**(4*a + 4*b*x)*b**3*d**4*x**3 + 29 16*e**(4*a + 4*b*x)*b**2*c**2*d**2 + 5832*e**(4*a + 4*b*x)*b**2*c*d**3*x + 2916*e**(4*a + 4*b*x)*b**2*d**4*x**2 - 5832*e**(4*a + 4*b*x)*b*c*d**3 - 5 832*e**(4*a + 4*b*x)*b*d**4*x + 5832*e**(4*a + 4*b*x)*d**4 - 243*e**(2*a + 2*b*x)*b**4*c**4 - 972*e**(2*a + 2*b*x)*b**4*c**3*d*x - 1458*e**(2*a + 2* b*x)*b**4*c**2*d**2*x**2 - 972*e**(2*a + 2*b*x)*b**4*c*d**3*x**3 - 243*e** (2*a + 2*b*x)*b**4*d**4*x**4 - 972*e**(2*a + 2*b*x)*b**3*c**3*d - 2916*e** (2*a + 2*b*x)*b**3*c**2*d**2*x - 2916*e**(2*a + 2*b*x)*b**3*c*d**3*x**2 - 972*e**(2*a + 2*b*x)*b**3*d**4*x**3 - 2916*e**(2*a + 2*b*x)*b**2*c**2*d...