Optimal. Leaf size=123 \[ -\frac{2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}+\frac{4 d (c+d x) \sinh (a+b x)}{3 b^2}+\frac{2 d^2 \cosh ^3(a+b x)}{27 b^3}-\frac{14 d^2 \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac{(c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
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Rubi [A] time = 0.131249, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3311, 3296, 2638, 2633} \[ -\frac{2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}+\frac{4 d (c+d x) \sinh (a+b x)}{3 b^2}+\frac{2 d^2 \cosh ^3(a+b x)}{27 b^3}-\frac{14 d^2 \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac{(c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3311
Rule 3296
Rule 2638
Rule 2633
Rubi steps
\begin{align*} \int (c+d x)^2 \sinh ^3(a+b x) \, dx &=\frac{(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac{2}{3} \int (c+d x)^2 \sinh (a+b x) \, dx+\frac{\left (2 d^2\right ) \int \sinh ^3(a+b x) \, dx}{9 b^2}\\ &=-\frac{2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac{(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}+\frac{(4 d) \int (c+d x) \cosh (a+b x) \, dx}{3 b}-\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{9 b^3}\\ &=-\frac{2 d^2 \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac{2 d^2 \cosh ^3(a+b x)}{27 b^3}+\frac{4 d (c+d x) \sinh (a+b x)}{3 b^2}+\frac{(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac{\left (4 d^2\right ) \int \sinh (a+b x) \, dx}{3 b^2}\\ &=-\frac{14 d^2 \cosh (a+b x)}{9 b^3}-\frac{2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac{2 d^2 \cosh ^3(a+b x)}{27 b^3}+\frac{4 d (c+d x) \sinh (a+b x)}{3 b^2}+\frac{(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac{2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.403865, size = 86, normalized size = 0.7 \[ \frac{-81 \cosh (a+b x) \left (b^2 (c+d x)^2+2 d^2\right )+\cosh (3 (a+b x)) \left (9 b^2 (c+d x)^2+2 d^2\right )-6 b d (c+d x) (\sinh (3 (a+b x))-27 \sinh (a+b x))}{108 b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 320, normalized size = 2.6 \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{2}}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}-{\frac{2\, \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) }{3}}-{\frac{ \left ( 2\,bx+2\,a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{9}}+{\frac{ \left ( 14\,bx+14\,a \right ) \sinh \left ( bx+a \right ) }{9}}+{\frac{2\, \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) }{27}}-{\frac{40\,\cosh \left ( bx+a \right ) }{27}} \right ) }-2\,{\frac{a{d}^{2}}{{b}^{2}} \left ( -2/3\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) +1/3\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) +{\frac{7\,\sinh \left ( bx+a \right ) }{9}}-1/9\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \right ) }+{\frac{{a}^{2}{d}^{2}\cosh \left ( bx+a \right ) }{{b}^{2}} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) }+2\,{\frac{cd}{b} \left ( -2/3\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) +1/3\, \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}\cosh \left ( bx+a \right ) +{\frac{7\,\sinh \left ( bx+a \right ) }{9}}-1/9\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \right ) }-2\,{\frac{cda \left ( -2/3+1/3\, \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \right ) \cosh \left ( bx+a \right ) }{b}}+{c}^{2} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.35653, size = 363, normalized size = 2.95 \begin{align*} \frac{1}{36} \, c 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^{2}{\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}{216} \, 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.62494, size = 466, normalized size = 3.79 \begin{align*} \frac{{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right )^{3} + 3 \,{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 6 \,{\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )^{3} - 81 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right ) + 18 \,{\left (9 \, b d^{2} x + 9 \, b c d -{\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )}{108 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.83939, size = 284, normalized size = 2.31 \begin{align*} \begin{cases} \frac{c^{2} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{2 c^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac{2 c d x \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{4 c d x \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac{d^{2} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{2 d^{2} x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac{14 c d \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac{4 c d \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} - \frac{14 d^{2} x \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac{4 d^{2} x \sinh{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac{14 d^{2} \sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{9 b^{3}} - \frac{40 d^{2} \cosh ^{3}{\left (a + b x \right )}}{27 b^{3}} & \text{for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) \sinh ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18285, size = 311, normalized size = 2.53 \begin{align*} \frac{{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 6 \, b d^{2} x - 6 \, b c d + 2 \, d^{2}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{3}} - \frac{3 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{8 \, b^{3}} - \frac{3 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + 2 \, d^{2}\right )} e^{\left (-b x - a\right )}}{8 \, b^{3}} + \frac{{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 6 \, b d^{2} x + 6 \, b c d + 2 \, d^{2}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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