Optimal. Leaf size=125 \[ \frac {1}{128} x \left (128 a^2+96 a b+35 b^2\right )+\frac {b (96 a+163 b) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac {b (160 a+93 b) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac {b^2 \sinh (c+d x) \cosh ^7(c+d x)}{8 d}-\frac {25 b^2 \sinh (c+d x) \cosh ^5(c+d x)}{48 d} \]
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Rubi [A] time = 0.16, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3209, 1157, 1814, 385, 206} \[ \frac {1}{128} x \left (128 a^2+96 a b+35 b^2\right )+\frac {b (96 a+163 b) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac {b (160 a+93 b) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac {b^2 \sinh (c+d x) \cosh ^7(c+d x)}{8 d}-\frac {25 b^2 \sinh (c+d x) \cosh ^5(c+d x)}{48 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 1157
Rule 1814
Rule 3209
Rubi steps
\begin {align*} \int \left (a+b \sinh ^4(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-2 a x^2+(a+b) x^4\right )^2}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}-\frac {\operatorname {Subst}\left (\int \frac {-8 a^2+b^2+8 \left (3 a^2+b^2\right ) x^2-8 (3 a-b) (a+b) x^4+8 (a+b)^2 x^6}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {25 b^2 \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}+\frac {\operatorname {Subst}\left (\int \frac {48 a^2+19 b^2-96 \left (a^2-b^2\right ) x^2+48 (a+b)^2 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac {b (96 a+163 b) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}-\frac {25 b^2 \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}-\frac {\operatorname {Subst}\left (\int \frac {-3 \left (64 a^2-32 a b-29 b^2\right )+192 (a+b)^2 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{192 d}\\ &=-\frac {b (160 a+93 b) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {b (96 a+163 b) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}-\frac {25 b^2 \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}+\frac {\left (128 a^2+96 a b+35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{128 d}\\ &=\frac {1}{128} \left (128 a^2+96 a b+35 b^2\right ) x-\frac {b (160 a+93 b) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {b (96 a+163 b) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}-\frac {25 b^2 \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 92, normalized size = 0.74 \[ \frac {24 \left (128 a^2+96 a b+35 b^2\right ) (c+d x)-96 b (16 a+7 b) \sinh (2 (c+d x))+24 b (8 a+7 b) \sinh (4 (c+d x))-32 b^2 \sinh (6 (c+d x))+3 b^2 \sinh (8 (c+d x))}{3072 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 205, normalized size = 1.64 \[ \frac {3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, {\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} - 8 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + {\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} - 80 \, b^{2} \cosh \left (d x + c\right )^{3} + 12 \, {\left (8 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (128 \, a^{2} + 96 \, a b + 35 \, b^{2}\right )} d x + 3 \, {\left (b^{2} \cosh \left (d x + c\right )^{7} - 8 \, b^{2} \cosh \left (d x + c\right )^{5} + 4 \, {\left (8 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 8 \, {\left (16 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 183, normalized size = 1.46 \[ \frac {1}{128} \, {\left (128 \, a^{2} + 96 \, a b + 35 \, b^{2}\right )} x + \frac {b^{2} e^{\left (8 \, d x + 8 \, c\right )}}{2048 \, d} - \frac {b^{2} e^{\left (6 \, d x + 6 \, c\right )}}{192 \, d} + \frac {b^{2} e^{\left (-6 \, d x - 6 \, c\right )}}{192 \, d} - \frac {b^{2} e^{\left (-8 \, d x - 8 \, c\right )}}{2048 \, d} + \frac {{\left (8 \, a b + 7 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} - \frac {{\left (16 \, a b + 7 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d} + \frac {{\left (16 \, a b + 7 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{64 \, d} - \frac {{\left (8 \, a b + 7 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 111, normalized size = 0.89 \[ \frac {b^{2} \left (\left (\frac {\left (\sinh ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sinh ^{5}\left (d x +c \right )\right )}{48}+\frac {35 \left (\sinh ^{3}\left (d x +c \right )\right )}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )+2 a b \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{2} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 183, normalized size = 1.46 \[ \frac {1}{32} \, a b {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + a^{2} x - \frac {1}{6144} \, b^{2} {\left (\frac {{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {1680 \, {\left (d x + c\right )}}{d} - \frac {672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 108, normalized size = 0.86 \[ \frac {21\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-84\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-4\,b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {3\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}-192\,a\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+24\,a\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+384\,a^2\,d\,x+105\,b^2\,d\,x+288\,a\,b\,d\,x}{384\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.56, size = 332, normalized size = 2.66 \[ \begin {cases} a^{2} x + \frac {3 a b x \sinh ^{4}{\left (c + d x \right )}}{4} - \frac {3 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {3 a b x \cosh ^{4}{\left (c + d x \right )}}{4} + \frac {5 a b \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} - \frac {3 a b \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{4 d} + \frac {35 b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {35 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {105 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {35 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {35 b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {93 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} - \frac {511 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} + \frac {385 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} - \frac {35 b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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