Optimal. Leaf size=161 \[ \frac {\left (128 a^2+352 a b+193 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac {1}{256} x \left (128 a^2+160 a b+63 b^2\right )+\frac {b (160 a+513 b) \sinh (c+d x) \cosh ^5(c+d x)}{480 d}-\frac {b (416 a+447 b) \sinh (c+d x) \cosh ^3(c+d x)}{384 d}+\frac {b^2 \sinh (c+d x) \cosh ^9(c+d x)}{10 d}-\frac {41 b^2 \sinh (c+d x) \cosh ^7(c+d x)}{80 d} \]
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Rubi [A] time = 0.28, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3217, 1257, 1814, 1157, 385, 206} \[ \frac {\left (128 a^2+352 a b+193 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac {1}{256} x \left (128 a^2+160 a b+63 b^2\right )+\frac {b (160 a+513 b) \sinh (c+d x) \cosh ^5(c+d x)}{480 d}-\frac {b (416 a+447 b) \sinh (c+d x) \cosh ^3(c+d x)}{384 d}+\frac {b^2 \sinh (c+d x) \cosh ^9(c+d x)}{10 d}-\frac {41 b^2 \sinh (c+d x) \cosh ^7(c+d x)}{80 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 1157
Rule 1257
Rule 1814
Rule 3217
Rubi steps
\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a-2 a x^2+(a+b) x^4\right )^2}{\left (1-x^2\right )^6} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^2 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}+\frac {\operatorname {Subst}\left (\int \frac {-b^2+10 \left (a^2-b^2\right ) x^2-10 \left (3 a^2+b^2\right ) x^4+10 (3 a-b) (a+b) x^6-10 (a+b)^2 x^8}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{10 d}\\ &=-\frac {41 b^2 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^2 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}-\frac {\operatorname {Subst}\left (\int \frac {-33 b^2-80 \left (a^2+3 b^2\right ) x^2+160 \left (a^2-b^2\right ) x^4-80 (a+b)^2 x^6}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{80 d}\\ &=\frac {b (160 a+513 b) \cosh ^5(c+d x) \sinh (c+d x)}{480 d}-\frac {41 b^2 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^2 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}+\frac {\operatorname {Subst}\left (\int \frac {-5 b (32 a+63 b)+480 (a-3 b) (a+b) x^2-480 (a+b)^2 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{480 d}\\ &=-\frac {b (416 a+447 b) \cosh ^3(c+d x) \sinh (c+d x)}{384 d}+\frac {b (160 a+513 b) \cosh ^5(c+d x) \sinh (c+d x)}{480 d}-\frac {41 b^2 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^2 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}-\frac {\operatorname {Subst}\left (\int \frac {-15 b (96 a+65 b)-1920 (a+b)^2 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{1920 d}\\ &=\frac {\left (128 a^2+352 a b+193 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{256 d}-\frac {b (416 a+447 b) \cosh ^3(c+d x) \sinh (c+d x)}{384 d}+\frac {b (160 a+513 b) \cosh ^5(c+d x) \sinh (c+d x)}{480 d}-\frac {41 b^2 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^2 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}-\frac {\left (128 a^2+160 a b+63 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{256 d}\\ &=-\frac {1}{256} \left (128 a^2+160 a b+63 b^2\right ) x+\frac {\left (128 a^2+352 a b+193 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{256 d}-\frac {b (416 a+447 b) \cosh ^3(c+d x) \sinh (c+d x)}{384 d}+\frac {b (160 a+513 b) \cosh ^5(c+d x) \sinh (c+d x)}{480 d}-\frac {41 b^2 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^2 \cosh ^9(c+d x) \sinh (c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 139, normalized size = 0.86 \[ -\frac {-60 \left (128 a^2+240 a b+105 b^2\right ) \sinh (2 (c+d x))+15360 a^2 c+15360 a^2 d x-320 a b \sinh (6 (c+d x))+360 b (8 a+5 b) \sinh (4 (c+d x))+19200 a b c+19200 a b d x-450 b^2 \sinh (6 (c+d x))+75 b^2 \sinh (8 (c+d x))-6 b^2 \sinh (10 (c+d x))+7560 b^2 c+7560 b^2 d x}{30720 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 305, normalized size = 1.89 \[ \frac {15 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{9} + 30 \, {\left (6 \, b^{2} \cosh \left (d x + c\right )^{3} - 5 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 3 \, {\left (126 \, b^{2} \cosh \left (d x + c\right )^{5} - 350 \, b^{2} \cosh \left (d x + c\right )^{3} + 5 \, {\left (32 \, a b + 45 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 10 \, {\left (18 \, b^{2} \cosh \left (d x + c\right )^{7} - 105 \, b^{2} \cosh \left (d x + c\right )^{5} + 5 \, {\left (32 \, a b + 45 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 36 \, {\left (8 \, a b + 5 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 30 \, {\left (128 \, a^{2} + 160 \, a b + 63 \, b^{2}\right )} d x + 15 \, {\left (b^{2} \cosh \left (d x + c\right )^{9} - 10 \, b^{2} \cosh \left (d x + c\right )^{7} + {\left (32 \, a b + 45 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} - 24 \, {\left (8 \, a b + 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 2 \, {\left (128 \, a^{2} + 240 \, a b + 105 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{7680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 241, normalized size = 1.50 \[ -\frac {1}{256} \, {\left (128 \, a^{2} + 160 \, a b + 63 \, b^{2}\right )} x + \frac {b^{2} e^{\left (10 \, d x + 10 \, c\right )}}{10240 \, d} - \frac {5 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )}}{4096 \, d} + \frac {5 \, b^{2} e^{\left (-8 \, d x - 8 \, c\right )}}{4096 \, d} - \frac {b^{2} e^{\left (-10 \, d x - 10 \, c\right )}}{10240 \, d} + \frac {{\left (32 \, a b + 45 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{6144 \, d} - \frac {3 \, {\left (8 \, a b + 5 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{512 \, d} + \frac {{\left (128 \, a^{2} + 240 \, a b + 105 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{1024 \, d} - \frac {{\left (128 \, a^{2} + 240 \, a b + 105 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{1024 \, d} + \frac {3 \, {\left (8 \, a b + 5 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{512 \, d} - \frac {{\left (32 \, a b + 45 \, b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{6144 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 148, normalized size = 0.92 \[ \frac {b^{2} \left (\left (\frac {\left (\sinh ^{9}\left (d x +c \right )\right )}{10}-\frac {9 \left (\sinh ^{7}\left (d x +c \right )\right )}{80}+\frac {21 \left (\sinh ^{5}\left (d x +c \right )\right )}{160}-\frac {21 \left (\sinh ^{3}\left (d x +c \right )\right )}{128}+\frac {63 \sinh \left (d x +c \right )}{256}\right ) \cosh \left (d x +c \right )-\frac {63 d x}{256}-\frac {63 c}{256}\right )+2 a b \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 260, normalized size = 1.61 \[ -\frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{20480} \, b^{2} {\left (\frac {{\left (25 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 \, e^{\left (-4 \, d x - 4 \, c\right )} + 600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 2100 \, e^{\left (-8 \, d x - 8 \, c\right )} - 2\right )} e^{\left (10 \, d x + 10 \, c\right )}}{d} + \frac {5040 \, {\left (d x + c\right )}}{d} + \frac {2100 \, e^{\left (-2 \, d x - 2 \, c\right )} - 600 \, e^{\left (-4 \, d x - 4 \, c\right )} + 150 \, e^{\left (-6 \, d x - 6 \, c\right )} - 25 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d}\right )} - \frac {1}{192} \, a b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 149, normalized size = 0.93 \[ \frac {960\,a^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {1575\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{2}-225\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+\frac {225\,b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-\frac {75\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}+\frac {3\,b^2\,\mathrm {sinh}\left (10\,c+10\,d\,x\right )}{4}+1800\,a\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-360\,a\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+40\,a\,b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )-1920\,a^2\,d\,x-945\,b^2\,d\,x-2400\,a\,b\,d\,x}{3840\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.71, size = 484, normalized size = 3.01 \[ \begin {cases} \frac {a^{2} x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a^{2} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {5 a b x \sinh ^{6}{\left (c + d x \right )}}{8} - \frac {15 a b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {15 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {5 a b x \cosh ^{6}{\left (c + d x \right )}}{8} + \frac {11 a b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {5 a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 a b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{8 d} + \frac {63 b^{2} x \sinh ^{10}{\left (c + d x \right )}}{256} - \frac {315 b^{2} x \sinh ^{8}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{256} + \frac {315 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{128} - \frac {315 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{128} + \frac {315 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{8}{\left (c + d x \right )}}{256} - \frac {63 b^{2} x \cosh ^{10}{\left (c + d x \right )}}{256} + \frac {193 b^{2} \sinh ^{9}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{256 d} - \frac {237 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {21 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{10 d} - \frac {147 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {63 b^{2} \sinh {\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{256 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\relax (c )}\right )^{2} \sinh ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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