Optimal. Leaf size=193 \[ \frac{b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 d (a-b)^3 \sqrt{a+b \coth ^2(c+d x)-b}}+\frac{b (9 a-5 b) \coth (c+d x)}{15 a^2 d (a-b)^2 \left (a+b \coth ^2(c+d x)-b\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{a^{7/2} d}+\frac{b \coth (c+d x)}{5 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^{5/2}} \]
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Rubi [A] time = 0.192174, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4128, 414, 527, 12, 377, 206} \[ \frac{b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 d (a-b)^3 \sqrt{a+b \coth ^2(c+d x)-b}}+\frac{b (9 a-5 b) \coth (c+d x)}{15 a^2 d (a-b)^2 \left (a+b \coth ^2(c+d x)-b\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{a^{7/2} d}+\frac{b \coth (c+d x)}{5 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4128
Rule 414
Rule 527
Rule 12
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \text{csch}^2(c+d x)\right )^{7/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{7/2}} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac{b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{-5 a+b+4 b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{5/2}} \, dx,x,\coth (c+d x)\right )}{5 a (a-b) d}\\ &=\frac{b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac{(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{15 a^2-12 a b+5 b^2-2 (9 a-5 b) b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )^{3/2}} \, dx,x,\coth (c+d x)\right )}{15 a^2 (a-b)^2 d}\\ &=\frac{b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac{(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac{b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 (a-b)^3 d \sqrt{a-b+b \coth ^2(c+d x)}}-\frac{\operatorname{Subst}\left (\int -\frac{15 (a-b)^3}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{15 a^3 (a-b)^3 d}\\ &=\frac{b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac{(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac{b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 (a-b)^3 d \sqrt{a-b+b \coth ^2(c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{a^3 d}\\ &=\frac{b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac{(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac{b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 (a-b)^3 d \sqrt{a-b+b \coth ^2(c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{a^3 d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{a^{7/2} d}+\frac{b \coth (c+d x)}{5 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )^{5/2}}+\frac{(9 a-5 b) b \coth (c+d x)}{15 a^2 (a-b)^2 d \left (a-b+b \coth ^2(c+d x)\right )^{3/2}}+\frac{b \left (33 a^2-40 a b+15 b^2\right ) \coth (c+d x)}{15 a^3 (a-b)^3 d \sqrt{a-b+b \coth ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.41247, size = 234, normalized size = 1.21 \[ \frac{\text{csch}^7(c+d x) \left (\frac{b \cosh (c+d x) \left (a^2 \left (45 a^2-60 a b+23 b^2\right ) \cosh (4 (c+d x))-4 a \left (-135 a^2 b+45 a^3+117 a b^2-35 b^3\right ) \cosh (2 (c+d x))+709 a^2 b^2-480 a^3 b+135 a^4-460 a b^3+120 b^4\right ) (a \cosh (2 (c+d x))-a+2 b)}{15 a^3 (a-b)^3}+\frac{\sqrt{2} (a \cosh (2 (c+d x))-a+2 b)^{7/2} \log \left (\sqrt{a \cosh (2 (c+d x))-a+2 b}+\sqrt{2} \sqrt{a} \cosh (c+d x)\right )}{a^{7/2}}\right )}{16 d \left (a+b \text{csch}^2(c+d x)\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.14, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ) ^{-{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{csch}\left (d x + c\right )^{2} + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{csch}^{2}{\left (c + d x \right )}\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{csch}\left (d x + c\right )^{2} + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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