Optimal. Leaf size=288 \[ -\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/4} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/4} d}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]
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Rubi [A]
time = 0.38, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3294, 1192,
1180, 211, 214} \begin {gather*} -\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cosh (c+d x) \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+11 a+b\right )}{32 a d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 1180
Rule 1192
Rule 3294
Rubi steps
\begin {align*} \int \frac {\sinh ^3(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {1-x^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {-12 a b+10 a b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{16 a (a-b) b d}\\ &=-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {4 a (13 a-b) b^2-4 a b^2 (5 a+b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{128 a^2 (a-b)^2 b^2 d}\\ &=-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^2 d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^2 d}\\ &=-\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/4} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/4} d}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.89, size = 802, normalized size = 2.78 \begin {gather*} \frac {\frac {32 \cosh (c+d x) (-17 a-b+(5 a+b) \cosh (2 (c+d x)))}{a (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+\frac {512 (a-b) (-5 \cosh (c+d x)+\cosh (3 (c+d x)))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}+\frac {\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {5 a c+b c+5 a d x+b d x+10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-47 a c \text {$\#$1}^2+5 b c \text {$\#$1}^2-47 a d x \text {$\#$1}^2+5 b d x \text {$\#$1}^2-94 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+10 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+47 a c \text {$\#$1}^4-5 b c \text {$\#$1}^4+47 a d x \text {$\#$1}^4-5 b d x \text {$\#$1}^4+94 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-10 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-5 a c \text {$\#$1}^6-b c \text {$\#$1}^6-5 a d x \text {$\#$1}^6-b d x \text {$\#$1}^6-10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6-2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a}}{256 (a-b)^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(592\) vs.
\(2(236)=472\).
time = 10.73, size = 593, normalized size = 2.06 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 20961 vs.
\(2 (233) = 466\).
time = 0.74, size = 20961, normalized size = 72.78 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1580 vs.
\(2 (233) = 466\).
time = 0.75, size = 1580, normalized size = 5.49 \begin {gather*} \frac {\frac {{\left ({\left (20 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} + 29 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )}^{2} {\left | b \right |} - {\left (52 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b - 43 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{2} - 75 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{3} + 71 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{4} - 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{5}\right )} {\left | a^{3} - 2 \, a^{2} b + a b^{2} \right |} {\left | b \right |} + 2 \, {\left (16 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{7} b - 48 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b^{2} + 27 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{3} + 52 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{4} - 78 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{5} + 36 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{6} - 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{3} b - 2 \, a^{2} b^{2} + a b^{3} + \sqrt {{\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} + {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}^{2}}}{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}}\right )}{{\left (4 \, a^{8} b^{3} - 15 \, a^{7} b^{4} + 15 \, a^{6} b^{5} + 10 \, a^{5} b^{6} - 30 \, a^{4} b^{7} + 21 \, a^{3} b^{8} - 5 \, a^{2} b^{9}\right )} {\left | a^{3} - 2 \, a^{2} b + a b^{2} \right |}} - \frac {{\left ({\left (20 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} + 29 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )}^{2} {\left | b \right |} + {\left (52 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b - 43 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{2} - 75 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{3} + 71 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{4} - 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{5}\right )} {\left | a^{3} - 2 \, a^{2} b + a b^{2} \right |} {\left | b \right |} + 2 \, {\left (16 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{7} b - 48 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b^{2} + 27 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{3} + 52 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{4} - 78 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{5} + 36 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{6} - 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{3} b - 2 \, a^{2} b^{2} + a b^{3} - \sqrt {{\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} + {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}^{2}}}{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}}\right )}{{\left (4 \, a^{8} b^{3} - 15 \, a^{7} b^{4} + 15 \, a^{6} b^{5} + 10 \, a^{5} b^{6} - 30 \, a^{4} b^{7} + 21 \, a^{3} b^{8} - 5 \, a^{2} b^{9}\right )} {\left | a^{3} - 2 \, a^{2} b + a b^{2} \right |}} - \frac {4 \, {\left (5 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 84 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 12 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 144 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 480 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 48 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 1216 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 1152 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 64 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 8 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 16 \, a + 16 \, b\right )}^{2} {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )}}}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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