Optimal. Leaf size=182 \[ \frac{a^2 (c+d x)^3}{3 d}-\frac{4 a b d (c+d x) \sinh (e+f x)}{f^2}+\frac{2 a b (c+d x)^2 \cosh (e+f x)}{f}+\frac{4 a b d^2 \cosh (e+f x)}{f^3}-\frac{b^2 d (c+d x) \sinh ^2(e+f x)}{2 f^2}+\frac{b^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}-\frac{b^2 (c+d x)^3}{6 d}+\frac{b^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}-\frac{b^2 d^2 x}{4 f^2} \]
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Rubi [A] time = 0.202018, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {3317, 3296, 2638, 3311, 32, 2635, 8} \[ \frac{a^2 (c+d x)^3}{3 d}-\frac{4 a b d (c+d x) \sinh (e+f x)}{f^2}+\frac{2 a b (c+d x)^2 \cosh (e+f x)}{f}+\frac{4 a b d^2 \cosh (e+f x)}{f^3}-\frac{b^2 d (c+d x) \sinh ^2(e+f x)}{2 f^2}+\frac{b^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}-\frac{b^2 (c+d x)^3}{6 d}+\frac{b^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}-\frac{b^2 d^2 x}{4 f^2} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3296
Rule 2638
Rule 3311
Rule 32
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (c+d x)^2 (a+b \sinh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a b (c+d x)^2 \sinh (e+f x)+b^2 (c+d x)^2 \sinh ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}+(2 a b) \int (c+d x)^2 \sinh (e+f x) \, dx+b^2 \int (c+d x)^2 \sinh ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}+\frac{2 a b (c+d x)^2 \cosh (e+f x)}{f}+\frac{b^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac{b^2 d (c+d x) \sinh ^2(e+f x)}{2 f^2}-\frac{1}{2} b^2 \int (c+d x)^2 \, dx+\frac{\left (b^2 d^2\right ) \int \sinh ^2(e+f x) \, dx}{2 f^2}-\frac{(4 a b d) \int (c+d x) \cosh (e+f x) \, dx}{f}\\ &=\frac{a^2 (c+d x)^3}{3 d}-\frac{b^2 (c+d x)^3}{6 d}+\frac{2 a b (c+d x)^2 \cosh (e+f x)}{f}-\frac{4 a b d (c+d x) \sinh (e+f x)}{f^2}+\frac{b^2 d^2 \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac{b^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac{b^2 d (c+d x) \sinh ^2(e+f x)}{2 f^2}+\frac{\left (4 a b d^2\right ) \int \sinh (e+f x) \, dx}{f^2}-\frac{\left (b^2 d^2\right ) \int 1 \, dx}{4 f^2}\\ &=-\frac{b^2 d^2 x}{4 f^2}+\frac{a^2 (c+d x)^3}{3 d}-\frac{b^2 (c+d x)^3}{6 d}+\frac{4 a b d^2 \cosh (e+f x)}{f^3}+\frac{2 a b (c+d x)^2 \cosh (e+f x)}{f}-\frac{4 a b d (c+d x) \sinh (e+f x)}{f^2}+\frac{b^2 d^2 \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac{b^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac{b^2 d (c+d x) \sinh ^2(e+f x)}{2 f^2}\\ \end{align*}
Mathematica [A] time = 0.752984, size = 249, normalized size = 1.37 \[ \frac{24 a^2 c^2 f^3 x+24 a^2 c d f^3 x^2+8 a^2 d^2 f^3 x^3+48 a b \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \cosh (e+f x)-96 a b c d f \sinh (e+f x)-96 a b d^2 f x \sinh (e+f x)+6 b^2 c^2 f^2 \sinh (2 (e+f x))-12 b^2 c^2 f^3 x+12 b^2 c d f^2 x \sinh (2 (e+f x))-6 b^2 d f (c+d x) \cosh (2 (e+f x))-12 b^2 c d f^3 x^2+6 b^2 d^2 f^2 x^2 \sinh (2 (e+f x))+3 b^2 d^2 \sinh (2 (e+f x))-4 b^2 d^2 f^3 x^3}{24 f^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 535, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27115, size = 435, normalized size = 2.39 \begin{align*} \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 )} b^{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 )} b^{2} d^{2} - \frac{1}{8} \, b^{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 b 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 b 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 b c^{2} \cosh \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.47877, size = 541, normalized size = 2.97 \begin{align*} \frac{2 \,{\left (2 \, a^{2} - b^{2}\right )} d^{2} f^{3} x^{3} + 6 \,{\left (2 \, a^{2} - b^{2}\right )} c d f^{3} x^{2} + 6 \,{\left (2 \, a^{2} - b^{2}\right )} c^{2} f^{3} x - 3 \,{\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cosh \left (f x + e\right )^{2} - 3 \,{\left (b^{2} d^{2} f x + b^{2} c d f\right )} \sinh \left (f x + e\right )^{2} + 24 \,{\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2} + 2 \, a b d^{2}\right )} \cosh \left (f x + e\right ) - 3 \,{\left (16 \, a b d^{2} f x + 16 \, a b c d f -{\left (2 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} c d f^{2} x + 2 \, b^{2} c^{2} f^{2} + b^{2} d^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{12 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.35746, size = 456, normalized size = 2.51 \begin{align*} \begin{cases} a^{2} c^{2} x + a^{2} c d x^{2} + \frac{a^{2} d^{2} x^{3}}{3} + \frac{2 a b c^{2} \cosh{\left (e + f x \right )}}{f} + \frac{4 a b c d x \cosh{\left (e + f x \right )}}{f} - \frac{4 a b c d \sinh{\left (e + f x \right )}}{f^{2}} + \frac{2 a b d^{2} x^{2} \cosh{\left (e + f x \right )}}{f} - \frac{4 a b d^{2} x \sinh{\left (e + f x \right )}}{f^{2}} + \frac{4 a b d^{2} \cosh{\left (e + f x \right )}}{f^{3}} + \frac{b^{2} c^{2} x \sinh ^{2}{\left (e + f x \right )}}{2} - \frac{b^{2} c^{2} x \cosh ^{2}{\left (e + f x \right )}}{2} + \frac{b^{2} c^{2} \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{2 f} + \frac{b^{2} c d x^{2} \sinh ^{2}{\left (e + f x \right )}}{2} - \frac{b^{2} c d x^{2} \cosh ^{2}{\left (e + f x \right )}}{2} + \frac{b^{2} c d x \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{f} - \frac{b^{2} c d \sinh ^{2}{\left (e + f x \right )}}{2 f^{2}} + \frac{b^{2} d^{2} x^{3} \sinh ^{2}{\left (e + f x \right )}}{6} - \frac{b^{2} d^{2} x^{3} \cosh ^{2}{\left (e + f x \right )}}{6} + \frac{b^{2} d^{2} x^{2} \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{2 f} - \frac{b^{2} d^{2} x \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac{b^{2} d^{2} x \cosh ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac{b^{2} d^{2} \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{4 f^{3}} & \text{for}\: f \neq 0 \\\left (a + b \sinh{\left (e \right )}\right )^{2} \left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2962, size = 470, normalized size = 2.58 \begin{align*} \frac{1}{3} \, a^{2} d^{2} x^{3} - \frac{1}{6} \, b^{2} d^{2} x^{3} + a^{2} c d x^{2} - \frac{1}{2} \, b^{2} c d x^{2} + a^{2} c^{2} x - \frac{1}{2} \, b^{2} c^{2} x + \frac{{\left (2 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} c d f^{2} x + 2 \, b^{2} c^{2} f^{2} - 2 \, b^{2} d^{2} f x - 2 \, b^{2} c d f + b^{2} d^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{16 \, f^{3}} + \frac{{\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2} - 2 \, a b d^{2} f x - 2 \, a b c d f + 2 \, a b d^{2}\right )} e^{\left (f x + e\right )}}{f^{3}} + \frac{{\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2} + 2 \, a b d^{2} f x + 2 \, a b c d f + 2 \, a b d^{2}\right )} e^{\left (-f x - e\right )}}{f^{3}} - \frac{{\left (2 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} c d f^{2} x + 2 \, b^{2} c^{2} f^{2} + 2 \, b^{2} d^{2} f x + 2 \, b^{2} c d f + b^{2} d^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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