Optimal. Leaf size=304 \[ -\frac {2 b^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {a^{2/3}+b^{2/3}}}-\frac {2 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {2 \sqrt [3]{-1} b^{2/3} \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\frac {\coth (c+d x)}{a d} \]
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Rubi [A] time = 0.48, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3220, 3767, 8, 2660, 618, 204} \[ -\frac {2 b^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {a^{2/3}+b^{2/3}}}-\frac {2 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {2 \sqrt [3]{-1} b^{2/3} \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}-\frac {\coth (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 204
Rule 618
Rule 2660
Rule 3220
Rule 3767
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=-\int \left (-\frac {\text {csch}^2(c+d x)}{a}+\frac {b \sinh (c+d x)}{a \left (a+b \sinh ^3(c+d x)\right )}\right ) \, dx\\ &=\frac {\int \text {csch}^2(c+d x) \, dx}{a}-\frac {b \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx}{a}\\ &=\frac {(i b) \int \left (\frac {\sqrt [3]{-1}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right )}-\frac {(-1)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)\right )}-\frac {1}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)\right )}\right ) \, dx}{a}-\frac {i \operatorname {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{a d}\\ &=-\frac {\coth (c+d x)}{a d}-\frac {\left (i b^{2/3}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{4/3}}+\frac {\left (\sqrt [6]{-1} b^{2/3}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{4/3}}+\frac {\left ((-1)^{5/6} b^{2/3}\right ) \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 a^{4/3}}\\ &=-\frac {\coth (c+d x)}{a d}-\frac {\left (2 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}+\frac {\left (2 \sqrt [3]{-1} b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-2 \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}-\frac {\left (2 (-1)^{2/3} b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-2 (-1)^{2/3} \sqrt [3]{b} x+\sqrt [6]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}\\ &=-\frac {\coth (c+d x)}{a d}+\frac {\left (4 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{-1} \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}-\frac {\left (4 \sqrt [3]{-1} b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}+\frac {\left (4 (-1)^{2/3} b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \sqrt [3]{-1} \left (a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,-2 (-1)^{2/3} \sqrt [3]{b}+2 \sqrt [6]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{3 a^{4/3} d}\\ &=-\frac {2 \sqrt [3]{-1} b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} d}-\frac {2 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 b^{2/3} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 a^{4/3} \sqrt {a^{2/3}+b^{2/3}} d}-\frac {\coth (c+d x)}{a d}\\ \end {align*}
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Mathematica [C] time = 0.42, size = 230, normalized size = 0.76 \[ -\frac {2 b \text {RootSum}\left [\text {$\#$1}^6 b-3 \text {$\#$1}^4 b+8 \text {$\#$1}^3 a+3 \text {$\#$1}^2 b-b\& ,\frac {2 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\text {$\#$1}^2 c+\text {$\#$1}^2 d x-2 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )-c-d x}{\text {$\#$1}^4 b-2 \text {$\#$1}^2 b+4 \text {$\#$1} a+b}\& \right ]+3 \tanh \left (\frac {1}{2} (c+d x)\right )+3 \coth \left (\frac {1}{2} (c+d x)\right )}{6 a d} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right )^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 123, normalized size = 0.40 \[ -\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}-\frac {2 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d a}-\frac {1}{2 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )} \]
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 \[ -\frac {2}{a d e^{\left (2 \, d x + 2 \, c\right )} - a d} - 4 \, \int \frac {b e^{\left (4 \, d x + 4 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )}}{a b e^{\left (6 \, d x + 6 \, c\right )} - 3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 3 \, a b e^{\left (2 \, d x + 2 \, c\right )} - a b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 23.06, size = 1293, normalized size = 4.25 \[ \left (\sum _{k=1}^6\ln \left (-\frac {8192\,b^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}-65536\,a\,b^3-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^2\,a^3\,b^3\,d^2\,294912-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^3\,a^4\,b^3\,d^3\,221184-\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )\,a^2\,b^3\,d\,196608+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^4\,a^8\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,10616832+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^5\,a^9\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,7962624-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^3\,a^6\,b\,d^3\,1769472+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^4\,a^7\,b\,d^4\,2654208-{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^5\,a^8\,b\,d^5\,1990656+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^2\,a^4\,b^2\,d^2\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,2064384+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^3\,a^5\,b^2\,d^3\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,5529600+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^4\,a^6\,b^2\,d^4\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,7299072+{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )}^5\,a^7\,b^2\,d^5\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,9953280+\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )\,a^3\,b^2\,d\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )+d\,x}\,393216}{a^4\,b^5}\right )\,\mathrm {root}\left (729\,a^8\,b^2\,d^6\,z^6+729\,a^{10}\,d^6\,z^6+243\,a^6\,b^2\,d^4\,z^4-b^4,z,k\right )\right )+\frac {2}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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