Integrand size = 14, antiderivative size = 91 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=-\frac {24 d^3 (c+d x) \cosh (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \cosh (a+b x)}{b^2}+\frac {24 d^4 \sinh (a+b x)}{b^5}+\frac {12 d^2 (c+d x)^2 \sinh (a+b x)}{b^3}+\frac {(c+d x)^4 \sinh (a+b x)}{b} \]
-24*d^3*(d*x+c)*cosh(b*x+a)/b^4-4*d*(d*x+c)^3*cosh(b*x+a)/b^2+24*d^4*sinh( b*x+a)/b^5+12*d^2*(d*x+c)^2*sinh(b*x+a)/b^3+(d*x+c)^4*sinh(b*x+a)/b
Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\frac {-4 b d (c+d x) \left (6 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)+\left (24 d^4+12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4\right ) \sinh (a+b x)}{b^5} \]
(-4*b*d*(c + d*x)*(6*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] + (24*d^4 + 12*b ^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Sinh[a + b*x])/b^5
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.25, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3777, 26, 3042, 26, 3777, 3042, 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.
| \(\displaystyle \int (c+d x)^4 \cosh (a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^4 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}-\frac {4 i d \int -i (c+d x)^3 \sinh (a+b x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}-\frac {4 d \int (c+d x)^3 \sinh (a+b x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}-\frac {4 d \int -i (c+d x)^3 \sin (i a+i b x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \int (c+d x)^3 \sin (i a+i b x)dx}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \int (c+d x)^2 \cosh (a+b x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \int (c+d x)^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 i d \int -i (c+d x) \sinh (a+b x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 d \int (c+d x) \sinh (a+b x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}-\frac {2 d \int -i (c+d x) \sin (i a+i b x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \int (c+d x) \sin (i a+i b x)dx}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \cosh (a+b x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {(c+d x)^4 \sinh (a+b x)}{b}+\frac {4 i d \left (\frac {i (c+d x)^3 \cosh (a+b x)}{b}-\frac {3 i d \left (\frac {(c+d x)^2 \sinh (a+b x)}{b}+\frac {2 i d \left (\frac {i (c+d x) \cosh (a+b x)}{b}-\frac {i d \sinh (a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{b}\) |
((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
3.1.1.3.1 Defintions of rubi rules used
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]
Time = 5.56 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.58
| method | result | size |
| parallelrisch | \(\frac {12 d^{2} x b \left (\left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}+2 d^{2}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+2 \left (-\left (d x +c \right )^{4} b^{4}-12 d^{2} \left (d x +c \right )^{2} b^{2}-24 d^{4}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+8 d \left (\frac {d x}{2}+c \right ) \left (\left (x^{2} d^{2}+c d x +c^{2}\right ) b^{2}+6 d^{2}\right ) b}{b^{5} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\) | \(144\) |
| risch | \(\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}-4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x -12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}-12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}-4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}-24 b \,d^{4} x -24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{b x +a}}{2 b^{5}}-\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x +12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}+12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}+4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}+24 b \,d^{4} x +24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{-b x -a}}{2 b^{5}}\) | \(325\) |
| parts | \(\frac {\sinh \left (b x +a \right ) d^{4} x^{4}}{b}+\frac {4 \sinh \left (b x +a \right ) c \,d^{3} x^{3}}{b}+\frac {6 \sinh \left (b x +a \right ) c^{2} d^{2} x^{2}}{b}+\frac {4 \sinh \left (b x +a \right ) c^{3} d x}{b}+\frac {\sinh \left (b x +a \right ) c^{4}}{b}-\frac {4 d \left (\frac {d^{3} \left (\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \cosh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \cosh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \cosh \left (b x +a \right )}{b}+c^{3} \cosh \left (b x +a \right )\right )}{b^{2}}\) | \(394\) |
| meijerg | \(-\frac {16 i d^{4} \cosh \left (a \right ) \sqrt {\pi }\, \left (-\frac {i x b \left (\frac {5 x^{2} b^{2}}{2}+15\right ) \cosh \left (b x \right )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} x^{4} b^{4}+\frac {15}{2} x^{2} b^{2}+15\right ) \sinh \left (b x \right )}{10 \sqrt {\pi }}\right )}{b^{5}}-\frac {16 d^{4} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} x^{4} b^{4}+\frac {9}{2} x^{2} b^{2}+9\right ) \cosh \left (b x \right )}{6 \sqrt {\pi }}+\frac {x b \left (\frac {3 x^{2} b^{2}}{2}+9\right ) \sinh \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{5}}+\frac {32 d^{3} c \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 x^{2} b^{2}}{2}+3\right ) \cosh \left (b x \right )}{4 \sqrt {\pi }}+\frac {x b \left (\frac {x^{2} b^{2}}{2}+3\right ) \sinh \left (b x \right )}{4 \sqrt {\pi }}\right )}{b^{4}}-\frac {32 i d^{3} c \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {i x b \left (\frac {5 x^{2} b^{2}}{2}+15\right ) \cosh \left (b x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 x^{2} b^{2}}{2}+15\right ) \sinh \left (b x \right )}{20 \sqrt {\pi }}\right )}{b^{4}}+\frac {24 i d^{2} c^{2} \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {i x b \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 x^{2} b^{2}}{2}+3\right ) \sinh \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{3}}+\frac {24 d^{2} c^{2} \sinh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {x^{2} b^{2}}{2}+1\right ) \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{3}}-\frac {8 d \,c^{3} \cosh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {4 d \,c^{3} \sinh \left (a \right ) \left (\cosh \left (b x \right ) x b -\sinh \left (b x \right )\right )}{b^{2}}+\frac {c^{4} \cosh \left (a \right ) \sinh \left (b x \right )}{b}-\frac {c^{4} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (b x \right )}{\sqrt {\pi }}\right )}{b}\) | \(458\) |
| derivativedivides | \(\frac {\frac {d^{4} \left (\left (b x +a \right )^{4} \sinh \left (b x +a \right )-4 \left (b x +a \right )^{3} \cosh \left (b x +a \right )+12 \left (b x +a \right )^{2} \sinh \left (b x +a \right )-24 \left (b x +a \right ) \cosh \left (b x +a \right )+24 \sinh \left (b x +a \right )\right )}{b^{4}}+\frac {d^{4} a^{4} \sinh \left (b x +a \right )}{b^{4}}-\frac {4 d^{4} a \left (\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )\right )}{b^{4}}+\frac {4 d^{3} c \left (\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )\right )}{b^{3}}+\frac {6 d^{4} a^{2} \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{4}}+\frac {6 d^{2} c^{2} \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}-\frac {4 d^{4} a^{3} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{4}}+\frac {4 d \,c^{3} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {4 d^{3} a^{3} c \sinh \left (b x +a \right )}{b^{3}}+\frac {6 d^{2} a^{2} c^{2} \sinh \left (b x +a \right )}{b^{2}}-\frac {4 d a \,c^{3} \sinh \left (b x +a \right )}{b}+c^{4} \sinh \left (b x +a \right )-\frac {12 d^{3} a c \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{3}}+\frac {12 d^{3} a^{2} c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{3}}-\frac {12 d^{2} a \,c^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}}{b}\) | \(547\) |
| default | \(\frac {\frac {d^{4} \left (\left (b x +a \right )^{4} \sinh \left (b x +a \right )-4 \left (b x +a \right )^{3} \cosh \left (b x +a \right )+12 \left (b x +a \right )^{2} \sinh \left (b x +a \right )-24 \left (b x +a \right ) \cosh \left (b x +a \right )+24 \sinh \left (b x +a \right )\right )}{b^{4}}+\frac {d^{4} a^{4} \sinh \left (b x +a \right )}{b^{4}}-\frac {4 d^{4} a \left (\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )\right )}{b^{4}}+\frac {4 d^{3} c \left (\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )\right )}{b^{3}}+\frac {6 d^{4} a^{2} \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{4}}+\frac {6 d^{2} c^{2} \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}-\frac {4 d^{4} a^{3} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{4}}+\frac {4 d \,c^{3} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {4 d^{3} a^{3} c \sinh \left (b x +a \right )}{b^{3}}+\frac {6 d^{2} a^{2} c^{2} \sinh \left (b x +a \right )}{b^{2}}-\frac {4 d a \,c^{3} \sinh \left (b x +a \right )}{b}+c^{4} \sinh \left (b x +a \right )-\frac {12 d^{3} a c \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{3}}+\frac {12 d^{3} a^{2} c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{3}}-\frac {12 d^{2} a \,c^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}}{b}\) | \(547\) |
2*(6*d^2*x*b*((1/3*x^2*d^2+c*d*x+c^2)*b^2+2*d^2)*tanh(1/2*b*x+1/2*a)^2+(-( d*x+c)^4*b^4-12*d^2*(d*x+c)^2*b^2-24*d^4)*tanh(1/2*b*x+1/2*a)+4*d*(1/2*d*x +c)*((d^2*x^2+c*d*x+c^2)*b^2+6*d^2)*b)/b^5/(tanh(1/2*b*x+1/2*a)^2-1)
Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.88 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=-\frac {4 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d + 6 \, b c d^{3} + 3 \, {\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right ) - {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + b^{4} c^{4} + 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 6 \, {\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 4 \, {\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{b^{5}} \]
-(4*(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) - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + b^4*c^4 + 12*b^2*c^2*d^2 + 24*d^4 + 6*(b^4*c^2*d^2 + 2*b^2*d^4)*x^2 + 4*(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. 311 vs. \(2 (92) = 184\).
Time = 0.36 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.42 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\begin {cases} \frac {c^{4} \sinh {\left (a + b x \right )}}{b} + \frac {4 c^{3} d x \sinh {\left (a + b x \right )}}{b} + \frac {6 c^{2} d^{2} x^{2} \sinh {\left (a + b x \right )}}{b} + \frac {4 c d^{3} x^{3} \sinh {\left (a + b x \right )}}{b} + \frac {d^{4} x^{4} \sinh {\left (a + b x \right )}}{b} - \frac {4 c^{3} d \cosh {\left (a + b x \right )}}{b^{2}} - \frac {12 c^{2} d^{2} x \cosh {\left (a + b x \right )}}{b^{2}} - \frac {12 c d^{3} x^{2} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {4 d^{4} x^{3} \cosh {\left (a + b x \right )}}{b^{2}} + \frac {12 c^{2} d^{2} \sinh {\left (a + b x \right )}}{b^{3}} + \frac {24 c d^{3} x \sinh {\left (a + b x \right )}}{b^{3}} + \frac {12 d^{4} x^{2} \sinh {\left (a + b x \right )}}{b^{3}} - \frac {24 c d^{3} \cosh {\left (a + b x \right )}}{b^{4}} - \frac {24 d^{4} x \cosh {\left (a + b x \right )}}{b^{4}} + \frac {24 d^{4} \sinh {\left (a + b x \right )}}{b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \cosh {\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((c**4*sinh(a + b*x)/b + 4*c**3*d*x*sinh(a + b*x)/b + 6*c**2*d**2 *x**2*sinh(a + b*x)/b + 4*c*d**3*x**3*sinh(a + b*x)/b + d**4*x**4*sinh(a + b*x)/b - 4*c**3*d*cosh(a + b*x)/b**2 - 12*c**2*d**2*x*cosh(a + b*x)/b**2 - 12*c*d**3*x**2*cosh(a + b*x)/b**2 - 4*d**4*x**3*cosh(a + b*x)/b**2 + 12* c**2*d**2*sinh(a + b*x)/b**3 + 24*c*d**3*x*sinh(a + b*x)/b**3 + 12*d**4*x* *2*sinh(a + b*x)/b**3 - 24*c*d**3*cosh(a + b*x)/b**4 - 24*d**4*x*cosh(a + b*x)/b**4 + 24*d**4*sinh(a + b*x)/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), True))
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (91) = 182\).
Time = 0.20 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.58 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\frac {c^{4} e^{\left (b x + a\right )}}{2 \, b} + \frac {2 \, {\left (b x e^{a} - e^{a}\right )} c^{3} d e^{\left (b x\right )}}{b^{2}} - \frac {c^{4} e^{\left (-b x - a\right )}}{2 \, b} - \frac {2 \, {\left (b x + 1\right )} c^{3} d e^{\left (-b x - a\right )}}{b^{2}} + \frac {3 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c^{2} d^{2} e^{\left (b x\right )}}{b^{3}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} c^{2} d^{2} e^{\left (-b x - a\right )}}{b^{3}} + \frac {2 \, {\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} c d^{3} e^{\left (b x\right )}}{b^{4}} - \frac {2 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} c d^{3} e^{\left (-b x - a\right )}}{b^{4}} + \frac {{\left (b^{4} x^{4} e^{a} - 4 \, b^{3} x^{3} e^{a} + 12 \, b^{2} x^{2} e^{a} - 24 \, b x e^{a} + 24 \, e^{a}\right )} d^{4} e^{\left (b x\right )}}{2 \, b^{5}} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} d^{4} e^{\left (-b x - a\right )}}{2 \, b^{5}} \]
1/2*c^4*e^(b*x + a)/b + 2*(b*x*e^a - e^a)*c^3*d*e^(b*x)/b^2 - 1/2*c^4*e^(- b*x - a)/b - 2*(b*x + 1)*c^3*d*e^(-b*x - a)/b^2 + 3*(b^2*x^2*e^a - 2*b*x*e ^a + 2*e^a)*c^2*d^2*e^(b*x)/b^3 - 3*(b^2*x^2 + 2*b*x + 2)*c^2*d^2*e^(-b*x - a)/b^3 + 2*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*c*d^3*e^(b* x)/b^4 - 2*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*c*d^3*e^(-b*x - a)/b^4 + 1/2* (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)*d^4*e ^(b*x)/b^5 - 1/2*(b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*d^4*e^(- b*x - a)/b^5
Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (91) = 182\).
Time = 0.26 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.56 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{2 \, b^{5}} - \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{2 \, b^{5}} \]
1/2*(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 - 1/2*(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
Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.36 \[ \int (c+d x)^4 \cosh (a+b x) \, dx=\frac {\mathrm {sinh}\left (a+b\,x\right )\,\left (b^4\,c^4+12\,b^2\,c^2\,d^2+24\,d^4\right )}{b^5}-\frac {4\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^3\,d+6\,c\,d^3\right )}{b^4}-\frac {4\,d^4\,x^3\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}-\frac {12\,x\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d^2+2\,d^4\right )}{b^4}+\frac {d^4\,x^4\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {4\,x\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^3\,d+6\,c\,d^3\right )}{b^3}+\frac {6\,x^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d^2+2\,d^4\right )}{b^3}-\frac {12\,c\,d^3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{b^2}+\frac {4\,c\,d^3\,x^3\,\mathrm {sinh}\left (a+b\,x\right )}{b} \]
(sinh(a + b*x)*(24*d^4 + b^4*c^4 + 12*b^2*c^2*d^2))/b^5 - (4*cosh(a + b*x) *(6*c*d^3 + b^2*c^3*d))/b^4 - (4*d^4*x^3*cosh(a + b*x))/b^2 - (12*x*cosh(a + b*x)*(2*d^4 + b^2*c^2*d^2))/b^4 + (d^4*x^4*sinh(a + b*x))/b + (4*x*sinh (a + b*x)*(6*c*d^3 + b^2*c^3*d))/b^3 + (6*x^2*sinh(a + b*x)*(2*d^4 + b^2*c ^2*d^2))/b^3 - (12*c*d^3*x^2*cosh(a + b*x))/b^2 + (4*c*d^3*x^3*sinh(a + b* x))/b