Optimal. Leaf size=290 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {a^{2/3}+b^{2/3}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 \sqrt [3]{-1} \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3220, 2660, 618, 204} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {a^{2/3}+b^{2/3}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 \sqrt [3]{-1} \tan ^{-1}\left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 2660
Rule 3220
Rubi steps
\begin {align*} \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx &=-\left (i \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\right )\\ &=\frac {i \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+(-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\sqrt [6]{-1} \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}+\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {(-1)^{5/6} \int \frac {1}{\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\\ &=\frac {2 \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 \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\left (2 \sqrt [3]{-1}\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 \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\left (2 (-1)^{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 \sqrt [3]{a} \sqrt [3]{b} d}\\ &=-\frac {4 \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 \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\left (4 \sqrt [3]{-1}\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 \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\left (4 (-1)^{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 \sqrt [3]{a} \sqrt [3]{b} d}\\ &=\frac {2 \sqrt [3]{-1} \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 \sqrt [3]{a} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} \sqrt [3]{b} d}+\frac {2 \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 \sqrt [3]{a} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} \sqrt [3]{b} d}+\frac {2 \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 \sqrt [3]{a} \sqrt {a^{2/3}+b^{2/3}} \sqrt [3]{b} d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.25, size = 199, normalized size = 0.69 \[ \frac {\text {RootSum}\left [\text {$\#$1}^6 b-3 \text {$\#$1}^4 b+8 \text {$\#$1}^3 a+3 \text {$\#$1}^2 b-b\& ,\frac {2 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\text {$\#$1}^2 c+\text {$\#$1}^2 d x-2 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )-c-d x}{\text {$\#$1}^4 b-2 \text {$\#$1}^2 b+4 \text {$\#$1} a+b}\& \right ]}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.09, size = 82, normalized size = 0.28 \[ \frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 21.84, size = 857, normalized size = 2.96 \[ \sum _{k=1}^6\ln \left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\frac {\left (4\,a^4\,b\,d^4+a^5\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,16+a^3\,b^2\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,11\right )\,663552}{b^6}-\frac {\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (-a^4\,b\,d^5+a^5\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,4+a^3\,b^2\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,5\right )\,1990656}{b^5}\right )+\frac {\left (8\,a^4\,d^3+a^2\,b^2\,d^3-a^3\,b\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,25\right )\,221184}{b^6}\right )-\frac {a^2\,d^2\,\left (b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,7\right )\,294912}{b^6}\right )+\frac {a^2\,d\,\left (b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,2\right )\,196608}{b^7}\right )-\frac {a\,\left (8\,a-b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\right )\,8192}{b^7}\right )\,\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh {\left (c + d x \right )}}{a + b \sinh ^{3}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________