Optimal. Leaf size=225 \[ \frac {8 d^4 \sinh ^3(a+b x)}{81 b^5}+\frac {488 d^4 \sinh (a+b x)}{27 b^5}-\frac {8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac {160 d^3 (c+d x) \cosh (a+b x)}{9 b^4}+\frac {80 d^2 (c+d x)^2 \sinh (a+b x)}{9 b^3}+\frac {4 d^2 (c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{9 b^3}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}-\frac {8 d (c+d x)^3 \cosh (a+b x)}{3 b^2}+\frac {2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]
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Rubi [A] time = 0.28, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3311, 3296, 2637, 2633} \[ \frac {80 d^2 (c+d x)^2 \sinh (a+b x)}{9 b^3}-\frac {8 d^3 (c+d x) \cosh ^3(a+b x)}{27 b^4}-\frac {160 d^3 (c+d x) \cosh (a+b x)}{9 b^4}+\frac {4 d^2 (c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{9 b^3}-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}-\frac {8 d (c+d x)^3 \cosh (a+b x)}{3 b^2}+\frac {8 d^4 \sinh ^3(a+b x)}{81 b^5}+\frac {488 d^4 \sinh (a+b x)}{27 b^5}+\frac {2 (c+d x)^4 \sinh (a+b x)}{3 b}+\frac {(c+d x)^4 \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2637
Rule 3296
Rule 3311
Rubi steps
\begin {align*} \int (c+d x)^4 \cosh ^3(a+b x) \, dx &=-\frac {4 d (c+d x)^3 \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {2}{3} \int (c+d x)^4 \cosh (a+b x) \, dx+\frac {\left (4 d^2\right ) \int (c+d x)^2 \cosh ^3(a+b x) \, dx}{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 {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) \int (c+d x)^3 \sinh (a+b x) \, dx}{3 b}+\frac {\left (8 d^2\right ) \int (c+d x)^2 \cosh (a+b x) \, dx}{9 b^2}+\frac {\left (8 d^4\right ) \int \cosh ^3(a+b x) \, dx}{27 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 {8 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 {\left (8 d^2\right ) \int (c+d x)^2 \cosh (a+b x) \, dx}{b^2}-\frac {\left (16 d^3\right ) \int (c+d x) \sinh (a+b x) \, dx}{9 b^3}+\frac {\left (8 i d^4\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{27 b^5}\\ &=-\frac {16 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 {8 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}-\frac {\left (16 d^3\right ) \int (c+d x) \sinh (a+b x) \, dx}{b^3}+\frac {\left (16 d^4\right ) \int \cosh (a+b x) \, dx}{9 b^4}\\ &=-\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 {56 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}+\frac {\left (16 d^4\right ) \int \cosh (a+b x) \, dx}{b^4}\\ &=-\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}\\ \end {align*}
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Mathematica [A] time = 0.97, size = 385, normalized size = 1.71 \[ \frac {243 b^4 c^4 \sinh (a+b x)+27 b^4 c^4 \sinh (3 (a+b x))+972 b^4 c^3 d x \sinh (a+b x)+108 b^4 c^3 d x \sinh (3 (a+b x))+1458 b^4 c^2 d^2 x^2 \sinh (a+b x)+162 b^4 c^2 d^2 x^2 \sinh (3 (a+b x))+972 b^4 c d^3 x^3 \sinh (a+b x)+108 b^4 c d^3 x^3 \sinh (3 (a+b x))+243 b^4 d^4 x^4 \sinh (a+b x)+27 b^4 d^4 x^4 \sinh (3 (a+b x))+2916 b^2 c^2 d^2 \sinh (a+b x)+36 b^2 c^2 d^2 \sinh (3 (a+b x))+5832 b^2 c d^3 x \sinh (a+b x)+72 b^2 c d^3 x \sinh (3 (a+b x))-972 b d (c+d x) \cosh (a+b x) \left (b^2 (c+d x)^2+6 d^2\right )-12 b d (c+d x) \cosh (3 (a+b x)) \left (3 b^2 (c+d x)^2+2 d^2\right )+2916 b^2 d^4 x^2 \sinh (a+b x)+36 b^2 d^4 x^2 \sinh (3 (a+b x))+5832 d^4 \sinh (a+b x)+8 d^4 \sinh (3 (a+b x))}{324 b^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 528, normalized size = 2.35 \[ -\frac {12 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right )^{3} + 36 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d + 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \sinh \left (b x + a\right )^{3} + 972 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d + 6 \, b c d^{3} + 3 \, {\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \cosh \left (b x + a\right ) - 3 \, {\left (81 \, b^{4} d^{4} x^{4} + 324 \, b^{4} c d^{3} x^{3} + 81 \, b^{4} c^{4} + 972 \, b^{2} c^{2} d^{2} + 1944 \, d^{4} + 486 \, {\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} + 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d + 2 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{2} + 324 \, {\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{324 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 654, normalized size = 2.91 \[ \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} - 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x - 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} - 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} - 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{648 \, b^{5}} + \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{8 \, b^{5}} - \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{8 \, b^{5}} - \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 36 \, b^{3} d^{4} x^{3} + 108 \, b^{4} c^{3} d x + 108 \, b^{3} c d^{3} x^{2} + 27 \, b^{4} c^{4} + 108 \, b^{3} c^{2} d^{2} x + 36 \, b^{2} d^{4} x^{2} + 36 \, b^{3} c^{3} d + 72 \, b^{2} c d^{3} x + 36 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 8 \, d^{4}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{648 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 1139, normalized size = 5.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 644, normalized size = 2.86 \[ \frac {1}{18} \, c^{3} 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^{4} {\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}{36} \, c^{2} 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}{54} \, c 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 )} + \frac {1}{648} \, d^{4} {\left (\frac {{\left (27 \, b^{4} x^{4} e^{\left (3 \, a\right )} - 36 \, b^{3} x^{3} e^{\left (3 \, a\right )} + 36 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 24 \, b x e^{\left (3 \, a\right )} + 8 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{5}} + \frac {243 \, {\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 )} e^{\left (b x\right )}}{b^{5}} - \frac {243 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} e^{\left (-b x - a\right )}}{b^{5}} - \frac {{\left (27 \, b^{4} x^{4} + 36 \, b^{3} x^{3} + 36 \, b^{2} x^{2} + 24 \, b x + 8\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 532, normalized size = 2.36 \[ \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (27\,b^4\,c^4+252\,b^2\,c^2\,d^2+488\,d^4\right )}{27\,b^5}-\frac {2\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (27\,b^4\,c^4+360\,b^2\,c^2\,d^2+728\,d^4\right )}{81\,b^5}-\frac {4\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (21\,b^2\,c^3\,d+122\,c\,d^3\right )}{27\,b^4}+\frac {8\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (3\,b^2\,c^3\,d+20\,c\,d^3\right )}{9\,b^4}-\frac {28\,d^4\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{9\,b^2}-\frac {4\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (63\,b^2\,c^2\,d^2+122\,d^4\right )}{27\,b^4}-\frac {2\,d^4\,x^4\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b}-\frac {8\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (3\,b^2\,c^3\,d+20\,c\,d^3\right )}{9\,b^3}-\frac {4\,x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (9\,b^2\,c^2\,d^2+20\,d^4\right )}{9\,b^3}+\frac {2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (9\,b^2\,c^2\,d^2+14\,d^4\right )}{3\,b^3}-\frac {28\,c\,d^3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b^2}+\frac {d^4\,x^4\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {8\,d^4\,x^3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{3\,b^2}-\frac {8\,c\,d^3\,x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b}+\frac {8\,x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^2\,c^2\,d^2+20\,d^4\right )}{9\,b^4}+\frac {4\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (3\,b^2\,c^3\,d+14\,c\,d^3\right )}{3\,b^3}+\frac {4\,c\,d^3\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {8\,c\,d^3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.95, size = 772, normalized size = 3.43 \[ \begin {cases} - \frac {2 c^{4} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {c^{4} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {8 c^{3} d x \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {4 c^{3} d x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {4 c^{2} d^{2} x^{2} \sinh ^{3}{\left (a + b x \right )}}{b} + \frac {6 c^{2} d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {8 c d^{3} x^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {4 c d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 d^{4} x^{4} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{4} x^{4} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} + \frac {8 c^{3} d \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{2}} - \frac {28 c^{3} d \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {8 c^{2} d^{2} x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {28 c^{2} d^{2} x \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {8 c d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {28 c d^{3} x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {8 d^{4} x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{2}} - \frac {28 d^{4} x^{3} \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac {80 c^{2} d^{2} \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {28 c^{2} d^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {160 c d^{3} x \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {56 c d^{3} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {80 d^{4} x^{2} \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {28 d^{4} x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} + \frac {160 c d^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{4}} - \frac {488 c d^{3} \cosh ^{3}{\left (a + b x \right )}}{27 b^{4}} + \frac {160 d^{4} x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{4}} - \frac {488 d^{4} x \cosh ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {1456 d^{4} \sinh ^{3}{\left (a + b x \right )}}{81 b^{5}} + \frac {488 d^{4} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{27 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 ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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