Integrand size = 16, antiderivative size = 175 \[ \int (c+d x)^3 \sinh ^3(a+b x) \, dx=-\frac {40 d^2 (c+d x) \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac {40 d^3 \sinh (a+b x)}{9 b^4}+\frac {2 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac {2 d^2 (c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d^3 \sinh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2} \] Output:
-40/9*d^2*(d*x+c)*cosh(b*x+a)/b^3-2/3*(d*x+c)^3*cosh(b*x+a)/b+40/9*d^3*sin h(b*x+a)/b^4+2*d*(d*x+c)^2*sinh(b*x+a)/b^2+2/9*d^2*(d*x+c)*cosh(b*x+a)*sin h(b*x+a)^2/b^3+1/3*(d*x+c)^3*cosh(b*x+a)*sinh(b*x+a)^2/b-2/27*d^3*sinh(b*x +a)^3/b^4-1/3*d*(d*x+c)^2*sinh(b*x+a)^3/b^2
Time = 0.59 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.73 \[ \int (c+d x)^3 \sinh ^3(a+b x) \, dx=\frac {-162 b (c+d x) \left (6 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)+6 b (c+d x) \left (2 d^2+3 b^2 (c+d x)^2\right ) \cosh (3 (a+b x))-4 d \left (-242 d^2-117 b^2 (c+d x)^2+\left (2 d^2+9 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{216 b^4} \] Input:
Integrate[(c + d*x)^3*Sinh[a + b*x]^3,x]
Output:
(-162*b*(c + d*x)*(6*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] + 6*b*(c + d*x)* (2*d^2 + 3*b^2*(c + d*x)^2)*Cosh[3*(a + b*x)] - 4*d*(-242*d^2 - 117*b^2*(c + d*x)^2 + (2*d^2 + 9*b^2*(c + d*x)^2)*Cosh[2*(a + b*x)])*Sinh[a + b*x])/ (216*b^4)
Result contains complex when optimal does not.
Time = 1.16 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.41, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {3042, 26, 3792, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 3791, 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.
| \(\displaystyle \int (c+d x)^3 \sinh ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i (c+d x)^3 \sin (i a+i b x)^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int (c+d x)^3 \sin (i a+i b x)^3dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle i \left (\frac {2 d^2 \int -i (c+d x) \sinh ^3(a+b x)dx}{3 b^2}+\frac {2}{3} \int i (c+d x)^3 \sinh (a+b x)dx+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {2 i d^2 \int (c+d x) \sinh ^3(a+b x)dx}{3 b^2}+\frac {2}{3} i \int (c+d x)^3 \sinh (a+b x)dx+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {2 i d^2 \int i (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} i \int -i (c+d x)^3 \sin (i a+i b x)dx+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} \int (c+d x)^3 \sin (i a+i b x)dx+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \int (c+d x)^2 \cosh (a+b x)dx}{b}\right )+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 i d \int -i (c+d x) \sinh (a+b x)dx}{b}\right )}{b}\right )+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 d \int (c+d x) \sinh (a+b x)dx}{b}\right )}{b}\right )+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 d \int -i (c+d x) \sin (i a+i b x)dx}{b}\right )}{b}\right )+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \int (c+d x) \sin (i a+i b x)dx}{b}\right )}{b}\right )+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \cosh (a+b x)dx}{b}\right )}{b}\right )}{b}\right )+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )}{b}\right )+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle i \left (\frac {2 d^2 \int (c+d x) \sin (i a+i b x)^3dx}{3 b^2}+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle i \left (\frac {2 d^2 \left (\frac {2}{3} \int i (c+d x) \sinh (a+b x)dx+\frac {i d \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{3 b^2}+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {2 d^2 \left (\frac {2}{3} i \int (c+d x) \sinh (a+b x)dx+\frac {i d \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{3 b^2}+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {2 d^2 \left (\frac {2}{3} i \int -i (c+d x) \sin (i a+i b x)dx+\frac {i d \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{3 b^2}+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {2 d^2 \left (\frac {2}{3} \int (c+d x) \sin (i a+i b x)dx+\frac {i d \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{3 b^2}+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle i \left (\frac {2 d^2 \left (\frac {2}{3} \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \cosh (a+b x)dx}{b}\right )+\frac {i d \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{3 b^2}+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {2 d^2 \left (\frac {2}{3} \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )+\frac {i d \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{3 b^2}+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle i \left (\frac {2 d^2 \left (\frac {2}{3} \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )+\frac {i d \sinh ^3(a+b x)}{9 b^2}-\frac {i (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )}{3 b^2}+\frac {i d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )-\frac {i (c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b}\right )\) |
Input:
Int[(c + d*x)^3*Sinh[a + b*x]^3,x]
Output:
I*(((-1/3*I)*(c + d*x)^3*Cosh[a + b*x]*Sinh[a + b*x]^2)/b + ((I/3)*d*(c + d*x)^2*Sinh[a + b*x]^3)/b^2 + (2*d^2*(((-1/3*I)*(c + d*x)*Cosh[a + b*x]*Si nh[a + b*x]^2)/b + ((I/9)*d*Sinh[a + b*x]^3)/b^2 + (2*((I*(c + d*x)*Cosh[a + b*x])/b - (I*d*Sinh[a + b*x])/b^2))/3))/(3*b^2) + (2*((I*(c + d*x)^3*Co sh[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))/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[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_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
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 = 0.75 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.81
| method | result | size |
| parallelrisch | \(\frac {3 b \left (\left (d x +c \right )^{2} b^{2}+\frac {2 d^{2}}{3}\right ) \left (d x +c \right ) \cosh \left (3 b x +3 a \right )-3 d \left (\left (d x +c \right )^{2} b^{2}+\frac {2 d^{2}}{9}\right ) \sinh \left (3 b x +3 a \right )-27 b \left (\left (d x +c \right )^{2} b^{2}+6 d^{2}\right ) \left (d x +c \right ) \cosh \left (b x +a \right )+81 \left (\left (d x +c \right )^{2} b^{2}+2 d^{2}\right ) d \sinh \left (b x +a \right )-24 b^{3} c^{3}-160 b c \,d^{2}}{36 b^{4}}\) | \(142\) |
| risch | \(\frac {\left (9 d^{3} x^{3} b^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x -9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}-18 b^{2} c \,d^{2} x -9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-2 d^{3}\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{4}}-\frac {3 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x -3 b^{2} d^{3} x^{2}+b^{3} c^{3}-6 b^{2} c \,d^{2} x -3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{b x +a}}{8 b^{4}}-\frac {3 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+6 d^{3}\right ) {\mathrm e}^{-b x -a}}{8 b^{4}}+\frac {\left (9 d^{3} x^{3} b^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x +9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}+18 b^{2} c \,d^{2} x +9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+2 d^{3}\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{4}}\) | \(415\) |
| orering | \(-\frac {20 d \left (9 d^{4} x^{4} b^{4}+36 b^{4} c \,d^{3} x^{3}+54 b^{4} c^{2} d^{2} x^{2}+36 b^{4} c^{3} d x +9 b^{4} c^{4}+22 b^{2} d^{4} x^{2}+44 b^{2} c \,d^{3} x +22 b^{2} c^{2} d^{2}-72 d^{4}\right ) \sinh \left (b x +a \right )^{3}}{27 b^{6} \left (d x +c \right )^{2}}+\frac {10 \left (3 d^{4} x^{4} b^{4}+12 b^{4} c \,d^{3} x^{3}+18 b^{4} c^{2} d^{2} x^{2}+12 b^{4} c^{3} d x +3 b^{4} c^{4}+2 b^{2} d^{4} x^{2}+4 b^{2} c \,d^{3} x +2 b^{2} c^{2} d^{2}-84 d^{4}\right ) \left (3 \left (d x +c \right )^{2} \sinh \left (b x +a \right )^{3} d +3 \left (d x +c \right )^{3} \sinh \left (b x +a \right )^{2} b \cosh \left (b x +a \right )\right )}{27 \left (d x +c \right )^{4} b^{6}}+\frac {4 d \left (9 d^{2} x^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}+50 d^{2}\right ) \left (6 \left (d x +c \right ) \sinh \left (b x +a \right )^{3} d^{2}+18 \left (d x +c \right )^{2} \sinh \left (b x +a \right )^{2} d b \cosh \left (b x +a \right )+6 \left (d x +c \right )^{3} \sinh \left (b x +a \right ) b^{2} \cosh \left (b x +a \right )^{2}+3 \left (d x +c \right )^{3} \sinh \left (b x +a \right )^{3} b^{2}\right )}{27 b^{6} \left (d x +c \right )^{3}}-\frac {\left (3 d^{2} x^{2} b^{2}+6 b^{2} c d x +3 b^{2} c^{2}+20 d^{2}\right ) \left (6 d^{3} \sinh \left (b x +a \right )^{3}+54 \left (d x +c \right ) \sinh \left (b x +a \right )^{2} d^{2} b \cosh \left (b x +a \right )+54 \left (d x +c \right )^{2} \sinh \left (b x +a \right ) d \,b^{2} \cosh \left (b x +a \right )^{2}+27 \left (d x +c \right )^{2} \sinh \left (b x +a \right )^{3} d \,b^{2}+6 \left (d x +c \right )^{3} b^{3} \cosh \left (b x +a \right )^{3}+21 \left (d x +c \right )^{3} \sinh \left (b x +a \right )^{2} b^{3} \cosh \left (b x +a \right )\right )}{27 \left (d x +c \right )^{2} b^{6}}\) | \(578\) |
| derivativedivides | \(\frac {\frac {d^{3} \left (-\frac {2 \left (b x +a \right )^{3} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )-\frac {40 \left (b x +a \right ) \cosh \left (b x +a \right )}{9}+\frac {40 \sinh \left (b x +a \right )}{9}-\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{9}-\frac {2 \sinh \left (b x +a \right )^{3}}{27}\right )}{b^{3}}-\frac {3 d^{3} a \left (-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {40 \cosh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{27}\right )}{b^{3}}+\frac {3 d^{2} c \left (-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {40 \cosh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{27}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b^{3}}-\frac {6 d^{2} a c \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b}-\frac {d^{3} a^{3} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}+c^{3} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}\) | \(634\) |
| default | \(\frac {\frac {d^{3} \left (-\frac {2 \left (b x +a \right )^{3} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )-\frac {40 \left (b x +a \right ) \cosh \left (b x +a \right )}{9}+\frac {40 \sinh \left (b x +a \right )}{9}-\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{9}-\frac {2 \sinh \left (b x +a \right )^{3}}{27}\right )}{b^{3}}-\frac {3 d^{3} a \left (-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {40 \cosh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{27}\right )}{b^{3}}+\frac {3 d^{2} c \left (-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {40 \cosh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )^{3}}{9}+\frac {2 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{27}\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b^{3}}-\frac {6 d^{2} a c \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b^{2}}+\frac {3 d \,c^{2} \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{2}}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\sinh \left (b x +a \right )^{3}}{9}\right )}{b}-\frac {d^{3} a^{3} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}+c^{3} \left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}\) | \(634\) |
Input:
int((d*x+c)^3*sinh(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/36*(3*b*((d*x+c)^2*b^2+2/3*d^2)*(d*x+c)*cosh(3*b*x+3*a)-3*d*((d*x+c)^2*b ^2+2/9*d^2)*sinh(3*b*x+3*a)-27*b*((d*x+c)^2*b^2+6*d^2)*(d*x+c)*cosh(b*x+a) +81*((d*x+c)^2*b^2+2*d^2)*d*sinh(b*x+a)-24*b^3*c^3-160*b*c*d^2)/b^4
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (161) = 322\).
Time = 0.11 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.97 \[ \int (c+d x)^3 \sinh ^3(a+b x) \, dx=\frac {3 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{3} + 9 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \sinh \left (b x + a\right )^{3} - 81 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} + 6 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) + 3 \, {\left (81 \, b^{2} d^{3} x^{2} + 162 \, b^{2} c d^{2} x + 81 \, b^{2} c^{2} d + 162 \, d^{3} - {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \] Input:
integrate((d*x+c)^3*sinh(b*x+a)^3,x, algorithm="fricas")
Output:
1/108*(3*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 3*b^3*c^3 + 2*b*c*d^2 + (9*b^3 *c^2*d + 2*b*d^3)*x)*cosh(b*x + a)^3 + 9*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 3*b^3*c^3 + 2*b*c*d^2 + (9*b^3*c^2*d + 2*b*d^3)*x)*cosh(b*x + a)*sinh(b* x + a)^2 - (9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 2*d^3)*sinh(b*x + a)^3 - 81*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 + 6*b*c*d^2 + 3*(b^3 *c^2*d + 2*b*d^3)*x)*cosh(b*x + a) + 3*(81*b^2*d^3*x^2 + 162*b^2*c*d^2*x + 81*b^2*c^2*d + 162*d^3 - (9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 2*d^3)*cosh(b*x + a)^2)*sinh(b*x + a))/b^4
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (173) = 346\).
Time = 0.47 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.83 \[ \int (c+d x)^3 \sinh ^3(a+b x) \, dx=\begin {cases} \frac {c^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {3 c^{2} d x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c^{2} d x \cosh ^{3}{\left (a + b x \right )}}{b} + \frac {3 c d^{2} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c d^{2} x^{2} \cosh ^{3}{\left (a + b x \right )}}{b} + \frac {d^{3} x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 d^{3} x^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac {7 c^{2} d \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 c^{2} d \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {14 c d^{2} x \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d^{2} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {7 d^{3} x^{2} \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 d^{3} x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {14 c d^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{3}} - \frac {40 c d^{2} \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {14 d^{3} x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{3}} - \frac {40 d^{3} x \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {122 d^{3} \sinh ^{3}{\left (a + b x \right )}}{27 b^{4}} + \frac {40 d^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sinh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**3*sinh(b*x+a)**3,x)
Output:
Piecewise((c**3*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*c**3*cosh(a + b*x)**3 /(3*b) + 3*c**2*d*x*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*c**2*d*x*cosh(a + b*x)**3/b + 3*c*d**2*x**2*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*c*d**2*x** 2*cosh(a + b*x)**3/b + d**3*x**3*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*d**3 *x**3*cosh(a + b*x)**3/(3*b) - 7*c**2*d*sinh(a + b*x)**3/(3*b**2) + 2*c**2 *d*sinh(a + b*x)*cosh(a + b*x)**2/b**2 - 14*c*d**2*x*sinh(a + b*x)**3/(3*b **2) + 4*c*d**2*x*sinh(a + b*x)*cosh(a + b*x)**2/b**2 - 7*d**3*x**2*sinh(a + b*x)**3/(3*b**2) + 2*d**3*x**2*sinh(a + b*x)*cosh(a + b*x)**2/b**2 + 14 *c*d**2*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**3) - 40*c*d**2*cosh(a + b*x)* *3/(9*b**3) + 14*d**3*x*sinh(a + b*x)**2*cosh(a + b*x)/(3*b**3) - 40*d**3* x*cosh(a + b*x)**3/(9*b**3) - 122*d**3*sinh(a + b*x)**3/(27*b**4) + 40*d** 3*sinh(a + b*x)*cosh(a + b*x)**2/(9*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d* x**2/2 + c*d**2*x**3 + d**3*x**4/4)*sinh(a)**3, True))
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (161) = 322\).
Time = 0.06 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.49 \[ \int (c+d x)^3 \sinh ^3(a+b x) \, dx=\frac {1}{24} \, c^{2} d {\left (\frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} - \frac {27 \, {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac {27 \, {\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} + \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac {1}{24} \, c^{3} {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} + \frac {1}{72} \, c d^{2} {\left (\frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{b^{3}} + \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{3}}\right )} + \frac {1}{216} \, d^{3} {\left (\frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{4}} - \frac {81 \, {\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{b^{4}} - \frac {81 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{b^{4}} + \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{4}}\right )} \] Input:
integrate((d*x+c)^3*sinh(b*x+a)^3,x, algorithm="maxima")
Output:
1/24*c^2*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^3*(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) + 1/72*c*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) + 1/216*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)
Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (161) = 322\).
Time = 0.13 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.37 \[ \int (c+d x)^3 \sinh ^3(a+b x) \, dx=\frac {{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x - 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} - 18 \, b^{2} c d^{2} x - 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 2 \, d^{3}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} - \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x - 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} - 6 \, b^{2} c d^{2} x - 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 6 \, d^{3}\right )} e^{\left (b x + a\right )}}{8 \, b^{4}} - \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} + \frac {{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x + 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 2 \, d^{3}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \] Input:
integrate((d*x+c)^3*sinh(b*x+a)^3,x, algorithm="giac")
Output:
1/216*(9*b^3*d^3*x^3 + 27*b^3*c*d^2*x^2 + 27*b^3*c^2*d*x - 9*b^2*d^3*x^2 + 9*b^3*c^3 - 18*b^2*c*d^2*x - 9*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 - 2*d^3) *e^(3*b*x + 3*a)/b^4 - 3/8*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x - 3*b^2*d^3*x^2 + b^3*c^3 - 6*b^2*c*d^2*x - 3*b^2*c^2*d + 6*b*d^3*x + 6*b* c*d^2 - 6*d^3)*e^(b*x + a)/b^4 - 3/8*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^ 3*c^2*d*x + 3*b^2*d^3*x^2 + b^3*c^3 + 6*b^2*c*d^2*x + 3*b^2*c^2*d + 6*b*d^ 3*x + 6*b*c*d^2 + 6*d^3)*e^(-b*x - a)/b^4 + 1/216*(9*b^3*d^3*x^3 + 27*b^3* c*d^2*x^2 + 27*b^3*c^2*d*x + 9*b^2*d^3*x^2 + 9*b^3*c^3 + 18*b^2*c*d^2*x + 9*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 + 2*d^3)*e^(-3*b*x - 3*a)/b^4
Time = 1.51 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.08 \[ \int (c+d x)^3 \sinh ^3(a+b x) \, dx=\frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (3\,b^2\,c^3+14\,c\,d^2\right )}{3\,b^3}-\frac {{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (63\,b^2\,c^2\,d+122\,d^3\right )}{27\,b^4}-\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (3\,b^2\,c^3+20\,c\,d^2\right )}{9\,b^3}+\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^4}-\frac {2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^3}-\frac {2\,d^3\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b}-\frac {7\,d^3\,x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b^2}-\frac {14\,c\,d^2\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b^2}+\frac {x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^2\,c^2\,d+14\,d^3\right )}{3\,b^3}-\frac {2\,c\,d^2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{b}+\frac {d^3\,x^3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b}+\frac {2\,d^3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}+\frac {3\,c\,d^2\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b}+\frac {4\,c\,d^2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2} \] Input:
int(sinh(a + b*x)^3*(c + d*x)^3,x)
Output:
(cosh(a + b*x)*sinh(a + b*x)^2*(14*c*d^2 + 3*b^2*c^3))/(3*b^3) - (sinh(a + b*x)^3*(122*d^3 + 63*b^2*c^2*d))/(27*b^4) - (2*cosh(a + b*x)^3*(20*c*d^2 + 3*b^2*c^3))/(9*b^3) + (2*cosh(a + b*x)^2*sinh(a + b*x)*(20*d^3 + 9*b^2*c ^2*d))/(9*b^4) - (2*x*cosh(a + b*x)^3*(20*d^3 + 9*b^2*c^2*d))/(9*b^3) - (2 *d^3*x^3*cosh(a + b*x)^3)/(3*b) - (7*d^3*x^2*sinh(a + b*x)^3)/(3*b^2) - (1 4*c*d^2*x*sinh(a + b*x)^3)/(3*b^2) + (x*cosh(a + b*x)*sinh(a + b*x)^2*(14* d^3 + 9*b^2*c^2*d))/(3*b^3) - (2*c*d^2*x^2*cosh(a + b*x)^3)/b + (d^3*x^3*c osh(a + b*x)*sinh(a + b*x)^2)/b + (2*d^3*x^2*cosh(a + b*x)^2*sinh(a + b*x) )/b^2 + (3*c*d^2*x^2*cosh(a + b*x)*sinh(a + b*x)^2)/b + (4*c*d^2*x*cosh(a + b*x)^2*sinh(a + b*x))/b^2
Time = 0.26 (sec) , antiderivative size = 678, normalized size of antiderivative = 3.87 \[ \int (c+d x)^3 \sinh ^3(a+b x) \, dx=\frac {9 e^{6 b x +6 a} b^{3} d^{3} x^{3}-9 e^{6 b x +6 a} b^{2} c^{2} d -9 e^{6 b x +6 a} b^{2} d^{3} x^{2}+6 e^{6 b x +6 a} b c \,d^{2}+6 e^{6 b x +6 a} b \,d^{3} x -81 e^{2 b x +2 a} b^{3} d^{3} x^{3}-243 e^{2 b x +2 a} b^{2} d^{3} x^{2}-243 e^{2 b x +2 a} b^{2} c^{2} d +2 d^{3}-81 e^{4 b x +4 a} b^{3} c^{3}+9 b^{3} d^{3} x^{3}+9 b^{2} d^{3} x^{2}-486 e^{2 b x +2 a} d^{3}-81 e^{4 b x +4 a} b^{3} d^{3} x^{3}+27 b^{3} c^{2} d x +9 b^{2} c^{2} d -486 e^{2 b x +2 a} b c \,d^{2}-486 e^{2 b x +2 a} b \,d^{3} x +243 e^{4 b x +4 a} b^{2} c^{2} d +243 e^{4 b x +4 a} b^{2} d^{3} x^{2}-486 e^{4 b x +4 a} b c \,d^{2}-486 e^{4 b x +4 a} b \,d^{3} x +27 b^{3} c \,d^{2} x^{2}+9 e^{6 b x +6 a} b^{3} c^{3}-81 e^{2 b x +2 a} b^{3} c^{3}+27 e^{6 b x +6 a} b^{3} c^{2} d x +27 e^{6 b x +6 a} b^{3} c \,d^{2} x^{2}-18 e^{6 b x +6 a} b^{2} c \,d^{2} x -243 e^{2 b x +2 a} b^{3} c \,d^{2} x^{2}-486 e^{2 b x +2 a} b^{2} c \,d^{2} x +6 b c \,d^{2}+6 b \,d^{3} x -243 e^{4 b x +4 a} b^{3} c^{2} d x -243 e^{4 b x +4 a} b^{3} c \,d^{2} x^{2}+486 e^{4 b x +4 a} b^{2} c \,d^{2} x +9 b^{3} c^{3}+18 b^{2} c \,d^{2} x -243 e^{2 b x +2 a} b^{3} c^{2} d x -2 e^{6 b x +6 a} d^{3}+486 e^{4 b x +4 a} d^{3}}{216 e^{3 b x +3 a} b^{4}} \] Input:
int((d*x+c)^3*sinh(b*x+a)^3,x)
Output:
(9*e**(6*a + 6*b*x)*b**3*c**3 + 27*e**(6*a + 6*b*x)*b**3*c**2*d*x + 27*e** (6*a + 6*b*x)*b**3*c*d**2*x**2 + 9*e**(6*a + 6*b*x)*b**3*d**3*x**3 - 9*e** (6*a + 6*b*x)*b**2*c**2*d - 18*e**(6*a + 6*b*x)*b**2*c*d**2*x - 9*e**(6*a + 6*b*x)*b**2*d**3*x**2 + 6*e**(6*a + 6*b*x)*b*c*d**2 + 6*e**(6*a + 6*b*x) *b*d**3*x - 2*e**(6*a + 6*b*x)*d**3 - 81*e**(4*a + 4*b*x)*b**3*c**3 - 243* e**(4*a + 4*b*x)*b**3*c**2*d*x - 243*e**(4*a + 4*b*x)*b**3*c*d**2*x**2 - 8 1*e**(4*a + 4*b*x)*b**3*d**3*x**3 + 243*e**(4*a + 4*b*x)*b**2*c**2*d + 486 *e**(4*a + 4*b*x)*b**2*c*d**2*x + 243*e**(4*a + 4*b*x)*b**2*d**3*x**2 - 48 6*e**(4*a + 4*b*x)*b*c*d**2 - 486*e**(4*a + 4*b*x)*b*d**3*x + 486*e**(4*a + 4*b*x)*d**3 - 81*e**(2*a + 2*b*x)*b**3*c**3 - 243*e**(2*a + 2*b*x)*b**3* c**2*d*x - 243*e**(2*a + 2*b*x)*b**3*c*d**2*x**2 - 81*e**(2*a + 2*b*x)*b** 3*d**3*x**3 - 243*e**(2*a + 2*b*x)*b**2*c**2*d - 486*e**(2*a + 2*b*x)*b**2 *c*d**2*x - 243*e**(2*a + 2*b*x)*b**2*d**3*x**2 - 486*e**(2*a + 2*b*x)*b*c *d**2 - 486*e**(2*a + 2*b*x)*b*d**3*x - 486*e**(2*a + 2*b*x)*d**3 + 9*b**3 *c**3 + 27*b**3*c**2*d*x + 27*b**3*c*d**2*x**2 + 9*b**3*d**3*x**3 + 9*b**2 *c**2*d + 18*b**2*c*d**2*x + 9*b**2*d**3*x**2 + 6*b*c*d**2 + 6*b*d**3*x + 2*d**3)/(216*e**(3*a + 3*b*x)*b**4)