Optimal. Leaf size=68 \[ \frac {16 e^{4 a} x \left (c x^n\right )^{4 b} \, _2F_1\left (4,\frac {1}{2} \left (4+\frac {1}{b n}\right );\frac {1}{2} \left (6+\frac {1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{4 b n+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5546, 5548, 263, 364} \[ \frac {16 e^{4 a} x \left (c x^n\right )^{4 b} \, _2F_1\left (4,\frac {1}{2} \left (4+\frac {1}{b n}\right );\frac {1}{2} \left (6+\frac {1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{4 b n+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 364
Rule 5546
Rule 5548
Rubi steps
\begin {align*} \int \text {csch}^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1}{n}} \text {csch}^4(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (16 e^{-4 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1-4 b+\frac {1}{n}}}{\left (1-e^{-2 a} x^{-2 b}\right )^4} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (16 e^{-4 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+4 b+\frac {1}{n}}}{\left (-e^{-2 a}+x^{2 b}\right )^4} \, dx,x,c x^n\right )}{n}\\ &=\frac {16 e^{4 a} x \left (c x^n\right )^{4 b} \, _2F_1\left (4,\frac {1}{2} \left (4+\frac {1}{b n}\right );\frac {1}{2} \left (6+\frac {1}{b n}\right );e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+4 b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 9.02, size = 200, normalized size = 2.94 \[ \frac {x \left (4 \left (4 b^2 n^2-1\right ) \, _2F_1\left (1,\frac {1}{2 b n};1+\frac {1}{2 b n};e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )+\text {csch}^3\left (a+b \log \left (c x^n\right )\right ) \left (\left (1-12 b^2 n^2\right ) \cosh \left (a+b \log \left (c x^n\right )\right )+\left (4 b^2 n^2-1\right ) \cosh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )-4 b n \sinh \left (a+b \log \left (c x^n\right )\right )\right )+4 e^{2 a} (2 b n-1) \left (c x^n\right )^{2 b} \, _2F_1\left (1,1+\frac {1}{2 b n};2+\frac {1}{2 b n};e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{24 b^3 n^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.81, size = 0, normalized size = 0.00 \[ \int \mathrm {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 16 \, {\left (4 \, b^{2} n^{2} - 1\right )} \int \frac {1}{96 \, {\left (b^{3} c^{b} n^{3} e^{\left (b \log \left (x^{n}\right ) + a\right )} + b^{3} n^{3}\right )}}\,{d x} - 16 \, {\left (4 \, b^{2} n^{2} - 1\right )} \int \frac {1}{96 \, {\left (b^{3} c^{b} n^{3} e^{\left (b \log \left (x^{n}\right ) + a\right )} - b^{3} n^{3}\right )}}\,{d x} - \frac {{\left (2 \, b c^{4 \, b} n + c^{4 \, b}\right )} x e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 2 \, {\left (6 \, b^{2} c^{2 \, b} n^{2} - b c^{2 \, b} n - c^{2 \, b}\right )} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - {\left (4 \, b^{2} n^{2} - 1\right )} x}{3 \, {\left (b^{3} c^{6 \, b} n^{3} e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b^{3} c^{4 \, b} n^{3} e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b^{3} c^{2 \, b} n^{3} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b^{3} n^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________