Optimal. Leaf size=146 \[ \frac {\left (48 a^2-208 a b+139 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac {\left (80 a^2-176 a b+93 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac {1}{128} x \left (48 a^2-80 a b+35 b^2\right )+\frac {b (16 a-13 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac {b^2 \sinh ^5(c+d x) \cosh ^3(c+d x)}{8 d} \]
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Rubi [A] time = 0.18, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3187, 463, 455, 1157, 385, 206} \[ \frac {\left (48 a^2-208 a b+139 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}-\frac {\left (80 a^2-176 a b+93 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac {1}{128} x \left (48 a^2-80 a b+35 b^2\right )+\frac {b (16 a-13 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac {b^2 \sinh ^5(c+d x) \cosh ^3(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 455
Rule 463
Rule 1157
Rule 3187
Rubi steps
\begin {align*} \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a-(a-b) x^2\right )^2}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}-\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (-8 a^2+5 b^2+8 (a-b)^2 x^2\right )}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {(16 a-13 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}-\frac {\operatorname {Subst}\left (\int \frac {(16 a-13 b) b+6 (16 a-13 b) b x^2-48 (a-b)^2 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac {\left (48 a^2-208 a b+139 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {(16 a-13 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}+\frac {\operatorname {Subst}\left (\int \frac {-3 \left (16 a^2-48 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 {\left (80 a^2-176 a b+93 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {\left (48 a^2-208 a b+139 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {(16 a-13 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}+\frac {\left (48 a^2-80 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 (48 a^2-80 a b+35 b^2\right ) x-\frac {\left (80 a^2-176 a b+93 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {\left (48 a^2-208 a b+139 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {(16 a-13 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b^2 \cosh ^3(c+d x) \sinh ^5(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 133, normalized size = 0.91 \[ \frac {-96 \left (8 a^2-15 a b+7 b^2\right ) \sinh (2 (c+d x))+24 \left (4 a^2-12 a b+7 b^2\right ) \sinh (4 (c+d x))+1152 a^2 c+1152 a^2 d x+32 a b \sinh (6 (c+d x))-1920 a b c-1920 a b d x-32 b^2 \sinh (6 (c+d x))+3 b^2 \sinh (8 (c+d x))+840 b^2 c+840 b^2 d x}{3072 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 238, normalized size = 1.63 \[ \frac {3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, {\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} + 8 \, {\left (a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + {\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} + 80 \, {\left (a b - b^{2}\right )} \cosh \left (d x + c\right )^{3} + 12 \, {\left (4 \, a^{2} - 12 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} d x + 3 \, {\left (b^{2} \cosh \left (d x + c\right )^{7} + 8 \, {\left (a b - b^{2}\right )} \cosh \left (d x + c\right )^{5} + 4 \, {\left (4 \, a^{2} - 12 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 8 \, {\left (8 \, a^{2} - 15 \, a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 215, normalized size = 1.47 \[ \frac {1}{128} \, {\left (48 \, a^{2} - 80 \, a b + 35 \, b^{2}\right )} x + \frac {b^{2} e^{\left (8 \, d x + 8 \, c\right )}}{2048 \, d} - \frac {b^{2} e^{\left (-8 \, d x - 8 \, c\right )}}{2048 \, d} + \frac {{\left (a b - b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{192 \, d} + \frac {{\left (4 \, a^{2} - 12 \, a b + 7 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} - \frac {{\left (8 \, a^{2} - 15 \, a b + 7 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d} + \frac {{\left (8 \, a^{2} - 15 \, a b + 7 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{64 \, d} - \frac {{\left (4 \, a^{2} - 12 \, a b + 7 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} - \frac {{\left (a b - b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 150, normalized size = 1.03 \[ \frac {b^{2} \left (\left (\frac {\left (\sinh ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sinh ^{5}\left (d x +c \right )\right )}{48}+\frac {35 \left (\sinh ^{3}\left (d x +c \right )\right )}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )+2 a b \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+a^{2} \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 267, normalized size = 1.83 \[ \frac {1}{64} \, a^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{6144} \, b^{2} {\left (\frac {{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {1680 \, {\left (d x + c\right )}}{d} - \frac {672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} - \frac {1}{192} \, a b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 149, normalized size = 1.02 \[ \frac {12\,a^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-96\,a^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-84\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+21\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-4\,b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {3\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}+180\,a\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-36\,a\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+4\,a\,b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+144\,a^2\,d\,x+105\,b^2\,d\,x-240\,a\,b\,d\,x}{384\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.70, size = 490, normalized size = 3.36 \[ \begin {cases} \frac {3 a^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {5 a^{2} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {3 a^{2} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {5 a b x \sinh ^{6}{\left (c + d x \right )}}{8} - \frac {15 a b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {15 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {5 a b x \cosh ^{6}{\left (c + d x \right )}}{8} + \frac {11 a b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {5 a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 a b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{8 d} + \frac {35 b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {35 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {105 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {35 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {35 b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {93 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} - \frac {511 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} + \frac {385 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} - \frac {35 b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\relax (c )}\right )^{2} \sinh ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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