Optimal. Leaf size=297 \[ -\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}-\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt {3} b \sqrt [3]{b \coth ^2(c+d x)} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt {3} b \sqrt [3]{b \coth ^2(c+d x)} \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^2(c+d x)} \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {2}{3}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 297, 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, 3473, 3476, 329, 296, 634, 618, 204, 628, 206} \[ -\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}-\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt {3} b \sqrt [3]{b \coth ^2(c+d x)} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt {3} b \sqrt [3]{b \coth ^2(c+d x)} \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^2(c+d x)} \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {2}{3}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 206
Rule 296
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3473
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \left (b \coth ^2(c+d x)\right )^{4/3} \, dx &=\frac {\left (b \sqrt [3]{b \coth ^2(c+d x)}\right ) \int \coth ^{\frac {8}{3}}(c+d x) \, dx}{\coth ^{\frac {2}{3}}(c+d x)}\\ &=-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}+\frac {\left (b \sqrt [3]{b \coth ^2(c+d x)}\right ) \int \coth ^{\frac {2}{3}}(c+d x) \, dx}{\coth ^{\frac {2}{3}}(c+d x)}\\ &=-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}-\frac {\left (b \sqrt [3]{b \coth ^2(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^{2/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \coth ^{\frac {2}{3}}(c+d x)}\\ &=-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}-\frac {\left (3 b \sqrt [3]{b \coth ^2(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^4}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {2}{3}}(c+d x)}\\ &=-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}+\frac {\left (b \sqrt [3]{b \coth ^2(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\left (b \sqrt [3]{b \coth ^2(c+d x)}\right ) \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 \coth ^{\frac {2}{3}}(c+d x)}+\frac {\left (b \sqrt [3]{b \coth ^2(c+d x)}\right ) \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 \coth ^{\frac {2}{3}}(c+d x)}\\ &=\frac {b \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{d \coth ^{\frac {2}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}-\frac {\left (b \sqrt [3]{b \coth ^2(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\left (b \sqrt [3]{b \coth ^2(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\left (3 b \sqrt [3]{b \coth ^2(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 \coth ^{\frac {2}{3}}(c+d x)}-\frac {\left (3 b \sqrt [3]{b \coth ^2(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 \coth ^{\frac {2}{3}}(c+d x)}\\ &=\frac {b \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{d \coth ^{\frac {2}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}-\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {\left (3 b \sqrt [3]{b \coth ^2(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 \coth ^{\frac {2}{3}}(c+d x)}+\frac {\left (3 b \sqrt [3]{b \coth ^2(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 \coth ^{\frac {2}{3}}(c+d x)}\\ &=\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{2 d \coth ^{\frac {2}{3}}(c+d x)}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{2 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {b \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \sqrt [3]{b \coth ^2(c+d x)}}{d \coth ^{\frac {2}{3}}(c+d x)}-\frac {3 b \coth (c+d x) \sqrt [3]{b \coth ^2(c+d x)}}{5 d}-\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}+\frac {b \sqrt [3]{b \coth ^2(c+d x)} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {2}{3}}(c+d x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.35, size = 166, normalized size = 0.56 \[ -\frac {\left (b \coth ^2(c+d x)\right )^{4/3} \left (12 \coth ^{\frac {5}{3}}(c+d x)-20 \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )-5 \left (-\log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )+\log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )\right )\right )}{20 d \coth ^{\frac {8}{3}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 1994, normalized size = 6.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth \left (d x + c\right )^{2}\right )^{\frac {4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {4}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth \left (d x + c\right )^{2}\right )^{\frac {4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{4/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {4}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________