Optimal. Leaf size=289 \[ -\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\sqrt {3} \coth ^{\frac {4}{3}}(c+d x) \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\sqrt {3} \coth ^{\frac {4}{3}}(c+d x) \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3658, 3474, 3476, 329, 296, 634, 618, 204, 628, 206} \[ -\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\sqrt {3} \coth ^{\frac {4}{3}}(c+d x) \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\sqrt {3} \coth ^{\frac {4}{3}}(c+d x) \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 206
Rule 296
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3474
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{2/3}} \, dx &=\frac {\coth ^{\frac {4}{3}}(c+d x) \int \frac {1}{\coth ^{\frac {4}{3}}(c+d x)} \, dx}{\left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \int \coth ^{\frac {2}{3}}(c+d x) \, dx}{\left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {x^{2/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {x^4}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {-\frac {1}{2}-\frac {x}{2}}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {-\frac {1}{2}+\frac {x}{2}}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ &=-\frac {3 \coth (c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{d \left (b \coth ^2(c+d x)\right )^{2/3}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \left (b \coth ^2(c+d x)\right )^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.07, size = 41, normalized size = 0.14 \[ -\frac {3 \coth (c+d x) \, _2F_1\left (-\frac {1}{6},1;\frac {5}{6};\coth ^2(c+d x)\right )}{d \left (b \coth ^2(c+d x)\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 2066, normalized size = 7.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \coth \left (d x + c\right )^{2}\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \coth \left (d x + c\right )^{2}\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________