Optimal. Leaf size=250 \[ \frac {a^2 (c+d x)^4}{4 d}+\frac {12 a b d^2 (c+d x) \sinh (e+f x)}{f^3}-\frac {6 a b d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {2 a b (c+d x)^3 \sinh (e+f x)}{f}-\frac {12 a b d^3 \cosh (e+f x)}{f^4}+\frac {3 b^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {b^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac {b^2 (c+d x)^4}{8 d}-\frac {3 b^2 d^3 \cosh ^2(e+f x)}{8 f^4}+\frac {3 b^2 d^3 x^2}{8 f^2} \]
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Rubi [A] time = 0.28, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3317, 3296, 2638, 3311, 32, 3310} \[ \frac {a^2 (c+d x)^4}{4 d}+\frac {12 a b d^2 (c+d x) \sinh (e+f x)}{f^3}-\frac {6 a b d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {2 a b (c+d x)^3 \sinh (e+f x)}{f}-\frac {12 a b d^3 \cosh (e+f x)}{f^4}+\frac {3 b^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {b^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac {b^2 (c+d x)^4}{8 d}-\frac {3 b^2 d^3 \cosh ^2(e+f x)}{8 f^4}+\frac {3 b^2 d^3 x^2}{8 f^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2638
Rule 3296
Rule 3310
Rule 3311
Rule 3317
Rubi steps
\begin {align*} \int (c+d x)^3 (a+b \cosh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \cosh (e+f x)+b^2 (c+d x)^3 \cosh ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \cosh (e+f x) \, dx+b^2 \int (c+d x)^3 \cosh ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}-\frac {3 b^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {2 a b (c+d x)^3 \sinh (e+f x)}{f}+\frac {b^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac {1}{2} b^2 \int (c+d x)^3 \, dx+\frac {\left (3 b^2 d^2\right ) \int (c+d x) \cosh ^2(e+f x) \, dx}{2 f^2}-\frac {(6 a b d) \int (c+d x)^2 \sinh (e+f x) \, dx}{f}\\ &=\frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}-\frac {6 a b d (c+d x)^2 \cosh (e+f x)}{f^2}-\frac {3 b^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {3 b^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {2 a b (c+d x)^3 \sinh (e+f x)}{f}+\frac {3 b^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {b^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac {\left (12 a b d^2\right ) \int (c+d x) \cosh (e+f x) \, dx}{f^2}+\frac {\left (3 b^2 d^2\right ) \int (c+d x) \, dx}{4 f^2}\\ &=\frac {3 b^2 c d^2 x}{4 f^2}+\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}-\frac {6 a b d (c+d x)^2 \cosh (e+f x)}{f^2}-\frac {3 b^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {3 b^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {12 a b d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {2 a b (c+d x)^3 \sinh (e+f x)}{f}+\frac {3 b^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {b^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac {\left (12 a b d^3\right ) \int \sinh (e+f x) \, dx}{f^3}\\ &=\frac {3 b^2 c d^2 x}{4 f^2}+\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}+\frac {b^2 (c+d x)^4}{8 d}-\frac {12 a b d^3 \cosh (e+f x)}{f^4}-\frac {6 a b d (c+d x)^2 \cosh (e+f x)}{f^2}-\frac {3 b^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {3 b^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {12 a b d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {2 a b (c+d x)^3 \sinh (e+f x)}{f}+\frac {3 b^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {b^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 1.52, size = 232, normalized size = 0.93 \[ \frac {2 f \left (f^3 x \left (2 a^2+b^2\right ) \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+16 a b (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+6\right )\right ) \sinh (e+f x)+b^2 (c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2+3\right )\right ) \sinh (2 (e+f x))\right )-96 a b d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \cosh (e+f x)-3 b^2 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (2 f^2 x^2+1\right )\right ) \cosh (2 (e+f x))}{16 f^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 409, normalized size = 1.64 \[ \frac {2 \, {\left (2 \, a^{2} + b^{2}\right )} d^{3} f^{4} x^{4} + 8 \, {\left (2 \, a^{2} + b^{2}\right )} c d^{2} f^{4} x^{3} + 12 \, {\left (2 \, a^{2} + b^{2}\right )} c^{2} d f^{4} x^{2} + 8 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} f^{4} x - 3 \, {\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} + b^{2} d^{3}\right )} \cosh \left (f x + e\right )^{2} - 3 \, {\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} + b^{2} d^{3}\right )} \sinh \left (f x + e\right )^{2} - 96 \, {\left (a b d^{3} f^{2} x^{2} + 2 \, a b c d^{2} f^{2} x + a b c^{2} d f^{2} + 2 \, a b d^{3}\right )} \cosh \left (f x + e\right ) + 4 \, {\left (8 \, a b d^{3} f^{3} x^{3} + 24 \, a b c d^{2} f^{3} x^{2} + 8 \, a b c^{3} f^{3} + 48 \, a b c d^{2} f + 24 \, {\left (a b c^{2} d f^{3} + 2 \, a b d^{3} f\right )} x + {\left (2 \, b^{2} d^{3} f^{3} x^{3} + 6 \, b^{2} c d^{2} f^{3} x^{2} + 2 \, b^{2} c^{3} f^{3} + 3 \, b^{2} c d^{2} f + 3 \, {\left (2 \, b^{2} c^{2} d f^{3} + b^{2} d^{3} f\right )} x\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{16 \, f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 603, normalized size = 2.41 \[ \frac {1}{4} \, a^{2} d^{3} x^{4} + \frac {1}{8} \, b^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {1}{2} \, b^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + \frac {3}{4} \, b^{2} c^{2} d x^{2} + a^{2} c^{3} x + \frac {1}{2} \, b^{2} c^{3} x + \frac {{\left (4 \, b^{2} d^{3} f^{3} x^{3} + 12 \, b^{2} c d^{2} f^{3} x^{2} + 12 \, b^{2} c^{2} d f^{3} x - 6 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c^{3} f^{3} - 12 \, b^{2} c d^{2} f^{2} x - 6 \, b^{2} c^{2} d f^{2} + 6 \, b^{2} d^{3} f x + 6 \, b^{2} c d^{2} f - 3 \, b^{2} d^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{32 \, f^{4}} + \frac {{\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + 3 \, a b c^{2} d f^{3} x - 3 \, a b d^{3} f^{2} x^{2} + a b c^{3} f^{3} - 6 \, a b c d^{2} f^{2} x - 3 \, a b c^{2} d f^{2} + 6 \, a b d^{3} f x + 6 \, a b c d^{2} f - 6 \, a b d^{3}\right )} e^{\left (f x + e\right )}}{f^{4}} - \frac {{\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + 3 \, a b c^{2} d f^{3} x + 3 \, a b d^{3} f^{2} x^{2} + a b c^{3} f^{3} + 6 \, a b c d^{2} f^{2} x + 3 \, a b c^{2} d f^{2} + 6 \, a b d^{3} f x + 6 \, a b c d^{2} f + 6 \, a b d^{3}\right )} e^{\left (-f x - e\right )}}{f^{4}} - \frac {{\left (4 \, b^{2} d^{3} f^{3} x^{3} + 12 \, b^{2} c d^{2} f^{3} x^{2} + 12 \, b^{2} c^{2} d f^{3} x + 6 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c^{3} f^{3} + 12 \, b^{2} c d^{2} f^{2} x + 6 \, b^{2} c^{2} d f^{2} + 6 \, b^{2} d^{3} f x + 6 \, b^{2} c d^{2} f + 3 \, b^{2} d^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{32 \, f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1061, normalized size = 4.24 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 523, normalized size = 2.09 \[ \frac {1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + \frac {3}{16} \, {\left (4 \, x^{2} + \frac {{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} - \frac {{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} b^{2} c^{2} d + \frac {1}{16} \, {\left (8 \, x^{3} + \frac {3 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} - 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{3}} - \frac {3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{3}}\right )} b^{2} c d^{2} + \frac {1}{32} \, {\left (4 \, x^{4} + \frac {{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} - 6 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 6 \, f x e^{\left (2 \, e\right )} - 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{4}} - \frac {{\left (4 \, f^{3} x^{3} + 6 \, f^{2} x^{2} + 6 \, f x + 3\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{4}}\right )} b^{2} d^{3} + \frac {1}{8} \, b^{2} c^{3} {\left (4 \, x + \frac {e^{\left (2 \, f x + 2 \, e\right )}}{f} - \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c^{3} x + 3 \, a b c^{2} d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + 3 \, a b c d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + a b d^{3} {\left (\frac {{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} - \frac {{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac {2 \, a b c^{3} \sinh \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 481, normalized size = 1.92 \[ a^2\,c^3\,x+\frac {b^2\,c^3\,x}{2}+\frac {a^2\,d^3\,x^4}{4}+\frac {b^2\,d^3\,x^4}{8}+\frac {3\,a^2\,c^2\,d\,x^2}{2}+a^2\,c\,d^2\,x^3+\frac {3\,b^2\,c^2\,d\,x^2}{4}+\frac {b^2\,c\,d^2\,x^3}{2}-\frac {3\,b^2\,d^3\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{16\,f^4}+\frac {b^2\,c^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}-\frac {12\,a\,b\,d^3\,\mathrm {cosh}\left (e+f\,x\right )}{f^4}+\frac {2\,a\,b\,c^3\,\mathrm {sinh}\left (e+f\,x\right )}{f}-\frac {3\,b^2\,d^3\,x^2\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{8\,f^2}+\frac {b^2\,d^3\,x^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}-\frac {3\,b^2\,c^2\,d\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{8\,f^2}+\frac {3\,b^2\,c\,d^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{8\,f^3}+\frac {3\,b^2\,d^3\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{8\,f^3}-\frac {3\,b^2\,c\,d^2\,x\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{4\,f^2}+\frac {3\,b^2\,c^2\,d\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}-\frac {6\,a\,b\,c^2\,d\,\mathrm {cosh}\left (e+f\,x\right )}{f^2}+\frac {12\,a\,b\,c\,d^2\,\mathrm {sinh}\left (e+f\,x\right )}{f^3}+\frac {12\,a\,b\,d^3\,x\,\mathrm {sinh}\left (e+f\,x\right )}{f^3}+\frac {3\,b^2\,c\,d^2\,x^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}-\frac {6\,a\,b\,d^3\,x^2\,\mathrm {cosh}\left (e+f\,x\right )}{f^2}+\frac {2\,a\,b\,d^3\,x^3\,\mathrm {sinh}\left (e+f\,x\right )}{f}+\frac {6\,a\,b\,c\,d^2\,x^2\,\mathrm {sinh}\left (e+f\,x\right )}{f}-\frac {12\,a\,b\,c\,d^2\,x\,\mathrm {cosh}\left (e+f\,x\right )}{f^2}+\frac {6\,a\,b\,c^2\,d\,x\,\mathrm {sinh}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.61, size = 779, normalized size = 3.12 \[ \begin {cases} a^{2} c^{3} x + \frac {3 a^{2} c^{2} d x^{2}}{2} + a^{2} c d^{2} x^{3} + \frac {a^{2} d^{3} x^{4}}{4} + \frac {2 a b c^{3} \sinh {\left (e + f x \right )}}{f} + \frac {6 a b c^{2} d x \sinh {\left (e + f x \right )}}{f} - \frac {6 a b c^{2} d \cosh {\left (e + f x \right )}}{f^{2}} + \frac {6 a b c d^{2} x^{2} \sinh {\left (e + f x \right )}}{f} - \frac {12 a b c d^{2} x \cosh {\left (e + f x \right )}}{f^{2}} + \frac {12 a b c d^{2} \sinh {\left (e + f x \right )}}{f^{3}} + \frac {2 a b d^{3} x^{3} \sinh {\left (e + f x \right )}}{f} - \frac {6 a b d^{3} x^{2} \cosh {\left (e + f x \right )}}{f^{2}} + \frac {12 a b d^{3} x \sinh {\left (e + f x \right )}}{f^{3}} - \frac {12 a b d^{3} \cosh {\left (e + f x \right )}}{f^{4}} - \frac {b^{2} c^{3} x \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c^{3} x \cosh ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c^{3} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {3 b^{2} c^{2} d x^{2} \sinh ^{2}{\left (e + f x \right )}}{4} + \frac {3 b^{2} c^{2} d x^{2} \cosh ^{2}{\left (e + f x \right )}}{4} + \frac {3 b^{2} c^{2} d x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {3 b^{2} c^{2} d \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {b^{2} c d^{2} x^{3} \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c d^{2} x^{3} \cosh ^{2}{\left (e + f x \right )}}{2} + \frac {3 b^{2} c d^{2} x^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {3 b^{2} c d^{2} x \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {3 b^{2} c d^{2} x \cosh ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {3 b^{2} c d^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} - \frac {b^{2} d^{3} x^{4} \sinh ^{2}{\left (e + f x \right )}}{8} + \frac {b^{2} d^{3} x^{4} \cosh ^{2}{\left (e + f x \right )}}{8} + \frac {b^{2} d^{3} x^{3} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {3 b^{2} d^{3} x^{2} \sinh ^{2}{\left (e + f x \right )}}{8 f^{2}} - \frac {3 b^{2} d^{3} x^{2} \cosh ^{2}{\left (e + f x \right )}}{8 f^{2}} + \frac {3 b^{2} d^{3} x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} - \frac {3 b^{2} d^{3} \sinh ^{2}{\left (e + f x \right )}}{8 f^{4}} & \text {for}\: f \neq 0 \\\left (a + b \cosh {\relax (e )}\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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