Optimal. Leaf size=161 \[ -\frac {1}{256} \left (128 a^2+160 a b+63 b^2\right ) x+\frac {\left (128 a^2+352 a b+193 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{256 d}-\frac {b (416 a+447 b) \cosh ^3(c+d x) \sinh (c+d x)}{384 d}+\frac {b (160 a+513 b) \cosh ^5(c+d x) \sinh (c+d x)}{480 d}-\frac {41 b^2 \cosh ^7(c+d x) \sinh (c+d x)}{80 d}+\frac {b^2 \cosh ^9(c+d x) \sinh (c+d x)}{10 d} \]
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Rubi [A]
time = 0.19, 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 = {3296, 1271,
1828, 1171, 393, 212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 393
Rule 1171
Rule 1271
Rule 1828
Rule 3296
Rubi steps
\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx &=\frac {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 ) \text {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 = 0.27, size = 139, normalized size = 0.86 \begin {gather*} -\frac {15360 a^2 c+19200 a b c+7560 b^2 c+15360 a^2 d x+19200 a b d x+7560 b^2 d x-60 \left (128 a^2+240 a b+105 b^2\right ) \sinh (2 (c+d x))+360 b (8 a+5 b) \sinh (4 (c+d x))-320 a b \sinh (6 (c+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))}{30720 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.52, size = 130, normalized size = 0.81
method | result | size |
default | \(\frac {\left (-\frac {15}{64} b^{2}-\frac {3}{8} a b \right ) \sinh \left (4 d x +4 c \right )}{4 d}+\frac {\left (\frac {45}{512} b^{2}+\frac {1}{16} a b \right ) \sinh \left (6 d x +6 c \right )}{6 d}+\frac {\left (\frac {105}{256} b^{2}+\frac {15}{16} a b +\frac {1}{2} a^{2}\right ) \sinh \left (2 d x +2 c \right )}{2 d}-\frac {a^{2} x}{2}-\frac {63 b^{2} x}{256}-\frac {5 a b x}{8}-\frac {5 b^{2} \sinh \left (8 d x +8 c \right )}{2048 d}+\frac {b^{2} \sinh \left (10 d x +10 c \right )}{5120 d}\) | \(130\) |
risch | \(-\frac {63 b^{2} x}{256}-\frac {5 a b x}{8}-\frac {a^{2} x}{2}+\frac {b^{2} {\mathrm e}^{10 d x +10 c}}{10240 d}-\frac {5 b^{2} {\mathrm e}^{8 d x +8 c}}{4096 d}+\frac {15 b^{2} {\mathrm e}^{6 d x +6 c}}{2048 d}+\frac {b \,{\mathrm e}^{6 d x +6 c} a}{192 d}-\frac {3 \,{\mathrm e}^{4 d x +4 c} a b}{64 d}-\frac {15 \,{\mathrm e}^{4 d x +4 c} b^{2}}{512 d}+\frac {{\mathrm e}^{2 d x +2 c} a^{2}}{8 d}+\frac {15 \,{\mathrm e}^{2 d x +2 c} a b}{64 d}+\frac {105 \,{\mathrm e}^{2 d x +2 c} b^{2}}{1024 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{2}}{8 d}-\frac {15 \,{\mathrm e}^{-2 d x -2 c} a b}{64 d}-\frac {105 \,{\mathrm e}^{-2 d x -2 c} b^{2}}{1024 d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} a b}{64 d}+\frac {15 \,{\mathrm e}^{-4 d x -4 c} b^{2}}{512 d}-\frac {15 b^{2} {\mathrm e}^{-6 d x -6 c}}{2048 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c} a}{192 d}+\frac {5 b^{2} {\mathrm e}^{-8 d x -8 c}}{4096 d}-\frac {b^{2} {\mathrm e}^{-10 d x -10 c}}{10240 d}\) | \(319\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 260, normalized size = 1.61 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 305 vs.
\(2 (149) = 298\).
time = 0.43, size = 305, normalized size = 1.89 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 484 vs.
\(2 (155) = 310\).
time = 2.25, size = 484, normalized size = 3.01 \begin {gather*} \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}{\left (c \right )}\right )^{2} \sinh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 241, normalized size = 1.50 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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