Optimal. Leaf size=168 \[ -\frac {a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}-\frac {4 a^2 d (c+d x) \cosh (e+f x)}{f^2}+\frac {2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac {a^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac {a^2 (c+d x)^3}{2 d}+\frac {4 a^2 d^2 \sinh (e+f x)}{f^3}+\frac {a^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac {a^2 d^2 x}{4 f^2} \]
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Rubi [A] time = 0.19, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3317, 3296, 2637, 3311, 32, 2635, 8} \[ -\frac {a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}-\frac {4 a^2 d (c+d x) \cosh (e+f x)}{f^2}+\frac {2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac {a^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac {a^2 (c+d x)^3}{2 d}+\frac {4 a^2 d^2 \sinh (e+f x)}{f^3}+\frac {a^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac {a^2 d^2 x}{4 f^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 2637
Rule 3296
Rule 3311
Rule 3317
Rubi steps
\begin {align*} \int (c+d x)^2 (a+a \cosh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a^2 (c+d x)^2 \cosh (e+f x)+a^2 (c+d x)^2 \cosh ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}+a^2 \int (c+d x)^2 \cosh ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x)^2 \cosh (e+f x) \, dx\\ &=\frac {a^2 (c+d x)^3}{3 d}-\frac {a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}+\frac {2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac {a^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac {1}{2} a^2 \int (c+d x)^2 \, dx+\frac {\left (a^2 d^2\right ) \int \cosh ^2(e+f x) \, dx}{2 f^2}-\frac {\left (4 a^2 d\right ) \int (c+d x) \sinh (e+f x) \, dx}{f}\\ &=\frac {a^2 (c+d x)^3}{2 d}-\frac {4 a^2 d (c+d x) \cosh (e+f x)}{f^2}-\frac {a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}+\frac {2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac {a^2 d^2 \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac {\left (a^2 d^2\right ) \int 1 \, dx}{4 f^2}+\frac {\left (4 a^2 d^2\right ) \int \cosh (e+f x) \, dx}{f^2}\\ &=\frac {a^2 d^2 x}{4 f^2}+\frac {a^2 (c+d x)^3}{2 d}-\frac {4 a^2 d (c+d x) \cosh (e+f x)}{f^2}-\frac {a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}+\frac {4 a^2 d^2 \sinh (e+f x)}{f^3}+\frac {2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac {a^2 d^2 \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 192, normalized size = 1.14 \[ \frac {a^2 \left (16 c^2 f^2 \sinh (e+f x)+2 c^2 f^2 \sinh (2 (e+f x))+12 c^2 f^3 x+32 c d f^2 x \sinh (e+f x)+4 c d f^2 x \sinh (2 (e+f x))-32 d f (c+d x) \cosh (e+f x)-2 d f (c+d x) \cosh (2 (e+f x))+12 c d f^3 x^2+16 d^2 f^2 x^2 \sinh (e+f x)+2 d^2 f^2 x^2 \sinh (2 (e+f x))+32 d^2 \sinh (e+f x)+d^2 \sinh (2 (e+f x))+4 d^2 f^3 x^3\right )}{8 f^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 227, normalized size = 1.35 \[ \frac {2 \, a^{2} d^{2} f^{3} x^{3} + 6 \, a^{2} c d f^{3} x^{2} + 6 \, a^{2} c^{2} f^{3} x - {\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cosh \left (f x + e\right )^{2} - {\left (a^{2} d^{2} f x + a^{2} c d f\right )} \sinh \left (f x + e\right )^{2} - 16 \, {\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cosh \left (f x + e\right ) + {\left (8 \, a^{2} d^{2} f^{2} x^{2} + 16 \, a^{2} c d f^{2} x + 8 \, a^{2} c^{2} f^{2} + 16 \, a^{2} d^{2} + {\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} + a^{2} d^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{4 \, f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 333, normalized size = 1.98 \[ \frac {1}{2} \, a^{2} d^{2} x^{3} + \frac {3}{2} \, a^{2} c d x^{2} + \frac {3}{2} \, a^{2} c^{2} x + \frac {{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} - 2 \, a^{2} d^{2} f x - 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{16 \, f^{3}} + \frac {{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2} - 2 \, a^{2} d^{2} f x - 2 \, a^{2} c d f + 2 \, a^{2} d^{2}\right )} e^{\left (f x + e\right )}}{f^{3}} - \frac {{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2} + 2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f + 2 \, a^{2} d^{2}\right )} e^{\left (-f x - e\right )}}{f^{3}} - \frac {{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} + 2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 541, normalized size = 3.22 \[ \frac {\frac {d^{2} a^{2} \left (f x +e \right )^{3}}{3 f^{2}}+\frac {2 d^{2} a^{2} \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{2}}+\frac {d^{2} a^{2} \left (\frac {\left (f x +e \right )^{2} \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {\left (f x +e \right )^{3}}{6}-\frac {\left (f x +e \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{2}+\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{4}+\frac {f x}{4}+\frac {e}{4}\right )}{f^{2}}-\frac {d^{2} e \,a^{2} \left (f x +e \right )^{2}}{f^{2}}-\frac {4 d^{2} e \,a^{2} \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{2}}-\frac {2 d^{2} e \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (f x +e \right )\right )}{4}\right )}{f^{2}}+\frac {d^{2} e^{2} a^{2} \left (f x +e \right )}{f^{2}}+\frac {2 d^{2} e^{2} a^{2} \sinh \left (f x +e \right )}{f^{2}}+\frac {d^{2} e^{2} a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}+\frac {c d \,a^{2} \left (f x +e \right )^{2}}{f}+\frac {4 c d \,a^{2} \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f}+\frac {2 c d \,a^{2} \left (\frac {\left (f x +e \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (f x +e \right )\right )}{4}\right )}{f}-\frac {2 c d e \,a^{2} \left (f x +e \right )}{f}-\frac {4 c d e \,a^{2} \sinh \left (f x +e \right )}{f}-\frac {2 c d e \,a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+c^{2} a^{2} \left (f x +e \right )+2 c^{2} a^{2} \sinh \left (f x +e \right )+c^{2} a^{2} \left (\frac {\cosh \left (f x +e \right ) \sinh \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 327, normalized size = 1.95 \[ \frac {1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + \frac {1}{8} \, {\left (4 \, x^{2} + \frac {{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} - \frac {{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} a^{2} c d + \frac {1}{48} \, {\left (8 \, x^{3} + \frac {3 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} - 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{3}} - \frac {3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{3}}\right )} a^{2} d^{2} + \frac {1}{8} \, a^{2} c^{2} {\left (4 \, x + \frac {e^{\left (2 \, f x + 2 \, e\right )}}{f} - \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c^{2} x + 2 \, a^{2} c d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + a^{2} d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {2 \, a^{2} c^{2} \sinh \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 257, normalized size = 1.53 \[ \frac {16\,a^2\,d^2\,\mathrm {sinh}\left (e+f\,x\right )+\frac {a^2\,d^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{2}+8\,a^2\,c^2\,f^2\,\mathrm {sinh}\left (e+f\,x\right )+6\,a^2\,c^2\,f^3\,x+a^2\,c^2\,f^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+2\,a^2\,d^2\,f^3\,x^3-a^2\,c\,d\,f\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )-16\,a^2\,d^2\,f\,x\,\mathrm {cosh}\left (e+f\,x\right )+a^2\,d^2\,f^2\,x^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+6\,a^2\,c\,d\,f^3\,x^2-a^2\,d^2\,f\,x\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )-16\,a^2\,c\,d\,f\,\mathrm {cosh}\left (e+f\,x\right )+8\,a^2\,d^2\,f^2\,x^2\,\mathrm {sinh}\left (e+f\,x\right )+16\,a^2\,c\,d\,f^2\,x\,\mathrm {sinh}\left (e+f\,x\right )+2\,a^2\,c\,d\,f^2\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.70, size = 456, normalized size = 2.71 \[ \begin {cases} - \frac {a^{2} c^{2} x \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{2} x \cosh ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{2} x + \frac {a^{2} c^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} c^{2} \sinh {\left (e + f x \right )}}{f} - \frac {a^{2} c d x^{2} \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c d x^{2} \cosh ^{2}{\left (e + f x \right )}}{2} + a^{2} c d x^{2} + \frac {a^{2} c d x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{f} + \frac {4 a^{2} c d x \sinh {\left (e + f x \right )}}{f} - \frac {a^{2} c d \sinh ^{2}{\left (e + f x \right )}}{2 f^{2}} - \frac {4 a^{2} c d \cosh {\left (e + f x \right )}}{f^{2}} - \frac {a^{2} d^{2} x^{3} \sinh ^{2}{\left (e + f x \right )}}{6} + \frac {a^{2} d^{2} x^{3} \cosh ^{2}{\left (e + f x \right )}}{6} + \frac {a^{2} d^{2} x^{3}}{3} + \frac {a^{2} d^{2} x^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} d^{2} x^{2} \sinh {\left (e + f x \right )}}{f} - \frac {a^{2} d^{2} x \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {a^{2} d^{2} x \cosh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {4 a^{2} d^{2} x \cosh {\left (e + f x \right )}}{f^{2}} + \frac {a^{2} d^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} + \frac {4 a^{2} d^{2} \sinh {\left (e + f x \right )}}{f^{3}} & \text {for}\: f \neq 0 \\\left (a \cosh {\relax (e )} + a\right )^{2} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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