Optimal. Leaf size=217 \[ -\frac {\left (\sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{5/4} d}-\frac {\left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{5/4} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]
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Rubi [A]
time = 0.22, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3294, 1219,
1180, 211, 214} \begin {gather*} -\frac {\left (\sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} b^{5/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} b^{5/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 b d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 1180
Rule 1219
Rule 3294
Rubi steps
\begin {align*} \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {2 a (a-3 b)+2 a b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{8 a (a-b) b d}\\ &=\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\left (\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 )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b} d}-\frac {\left (a+\sqrt {a} \sqrt {b}-2 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 \sqrt {a} (a-b) \sqrt {b} d}\\ &=-\frac {\left (\sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{5/4} d}-\frac {\left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{5/4} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 (a-b) b 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 = 0.41, size = 597, normalized size = 2.75 \begin {gather*} \frac {\frac {32 \cosh (c+d x) (2 a+b-b \cosh (2 (c+d x)))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}+\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-b c-b d x-2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-4 a c \text {$\#$1}^2+11 b c \text {$\#$1}^2-4 a d x \text {$\#$1}^2+11 b d x \text {$\#$1}^2-8 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+22 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+4 a c \text {$\#$1}^4-11 b c \text {$\#$1}^4+4 a d x \text {$\#$1}^4-11 b d x \text {$\#$1}^4+8 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-22 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+b c \text {$\#$1}^6+b d x \text {$\#$1}^6+2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{32 (a-b) b 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. \(346\) vs.
\(2(167)=334\).
time = 5.13, size = 347, normalized size = 1.60
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {\left (-2 b +a \right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b \left (a -b \right )}+\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b \left (a -b \right )}-\frac {\left (3 a +2 b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b \left (a -b \right )}+\frac {a}{2 b \left (a -b \right )}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {a \left (-\frac {\left (-\sqrt {a b}-a +2 b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (\sqrt {a b}-a +2 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}\right )}{2 b \left (a -b \right )}}{d}\) | \(347\) |
default | \(\frac {\frac {-\frac {\left (-2 b +a \right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b \left (a -b \right )}+\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b \left (a -b \right )}-\frac {\left (3 a +2 b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 b \left (a -b \right )}+\frac {a}{2 b \left (a -b \right )}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {a \left (-\frac {\left (-\sqrt {a b}-a +2 b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (\sqrt {a b}-a +2 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}\right )}{2 b \left (a -b \right )}}{d}\) | \(347\) |
risch | \(\frac {{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{6 d x +6 c}+4 a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}+b \,{\mathrm e}^{2 d x +2 c}-b \right )}{2 b \left (a -b \right ) d \left (-b \,{\mathrm e}^{8 d x +8 c}+4 b \,{\mathrm e}^{6 d x +6 c}+16 a \,{\mathrm e}^{4 d x +4 c}-6 b \,{\mathrm e}^{4 d x +4 c}+4 b \,{\mathrm e}^{2 d x +2 c}-b \right )}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (65536 a^{5} b^{5} d^{4}-196608 a^{4} b^{6} d^{4}+196608 a^{3} b^{7} d^{4}-65536 a^{2} b^{8} d^{4}\right ) \textit {\_Z}^{4}+\left (-512 a^{3} d^{2} b^{3}+512 a^{2} b^{4} d^{2}+2048 a \,b^{5} d^{2}\right ) \textit {\_Z}^{2}-a^{2}+8 a b -16 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {16384 a^{4} b^{5} d^{3}}{a^{3}-9 a^{2} b +28 a \,b^{2}-32 b^{3}}-\frac {49152 a^{3} b^{6} d^{3}}{a^{3}-9 a^{2} b +28 a \,b^{2}-32 b^{3}}+\frac {49152 a^{2} b^{7} d^{3}}{a^{3}-9 a^{2} b +28 a \,b^{2}-32 b^{3}}-\frac {16384 a \,b^{8} d^{3}}{a^{3}-9 a^{2} b +28 a \,b^{2}-32 b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {32 a^{4} b d}{a^{3}-9 a^{2} b +28 a \,b^{2}-32 b^{3}}+\frac {256 a^{3} b^{2} d}{a^{3}-9 a^{2} b +28 a \,b^{2}-32 b^{3}}-\frac {800 a^{2} b^{3} d}{a^{3}-9 a^{2} b +28 a \,b^{2}-32 b^{3}}+\frac {832 a \,b^{4} d}{a^{3}-9 a^{2} b +28 a \,b^{2}-32 b^{3}}+\frac {256 b^{5} d}{a^{3}-9 a^{2} b +28 a \,b^{2}-32 b^{3}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\) | \(572\) |
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. 6250 vs.
\(2 (169) = 338\).
time = 0.56, size = 6250, normalized size = 28.80 \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. 984 vs.
\(2 (169) = 338\).
time = 0.69, size = 984, normalized size = 4.53 \begin {gather*} -\frac {\frac {{\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b\right )} {\left (a b - b^{2}\right )}^{2} {\left | b \right |} + {\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b - 11 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{2} - 8 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{3} + 15 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{4}\right )} {\left | -a b + b^{2} \right |} {\left | b \right |} - {\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{2} - 11 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{3} + 17 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{5} - 10 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{6}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b^{2} - b^{3} + \sqrt {{\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} {\left (a b^{2} - b^{3}\right )} + {\left (a b^{2} - b^{3}\right )}^{2}}}{a b^{2} - b^{3}}}}\right )}{{\left (4 \, a^{5} b^{4} - 7 \, a^{4} b^{5} - 3 \, a^{3} b^{6} + 11 \, a^{2} b^{7} - 5 \, a b^{8}\right )} {\left | -a b + b^{2} \right |}} - \frac {{\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b\right )} {\left (a b - b^{2}\right )}^{2} {\left | b \right |} - {\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b - 11 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{2} - 8 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{3} + 15 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{4}\right )} {\left | -a b + b^{2} \right |} {\left | b \right |} - {\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{2} - 11 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{3} + 17 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{5} - 10 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{6}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b^{2} - b^{3} - \sqrt {{\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} {\left (a b^{2} - b^{3}\right )} + {\left (a b^{2} - b^{3}\right )}^{2}}}{a b^{2} - b^{3}}}}\right )}{{\left (4 \, a^{5} b^{4} - 7 \, a^{4} b^{5} - 3 \, a^{3} b^{6} + 11 \, a^{2} b^{7} - 5 \, a b^{8}\right )} {\left | -a b + b^{2} \right |}} - \frac {4 \, {\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 4 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 4 \, b {\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 )} {\left (a b - b^{2}\right )}}}{8 \, 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 )}^5}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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