Optimal. Leaf size=208 \[ -\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left (8 a^2-40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3194, 89, 78, 51, 63, 208} \[ \frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\left (8 a^2-40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 89
Rule 208
Rule 3194
Rubi steps
\begin {align*} \int \frac {\coth ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(1+x)^2}{x^3 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (8 a-7 b)+2 a x}{x^2 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 a f}\\ &=-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{16 a^2 f}\\ &=\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{16 a^3 f}\\ &=\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{16 a^4 f}\\ &=\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\left (8 a^2-40 a b+35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^2(e+f x)}\right )}{8 a^4 b f}\\ &=-\frac {\left (8 a^2-40 a b+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{9/2} f}+\frac {8 a^2-40 a b+35 b^2}{24 a^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(8 a-7 b) \text {csch}^2(e+f x)}{8 a^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\text {csch}^4(e+f x)}{4 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {8 a^2-40 a b+35 b^2}{8 a^4 f \sqrt {a+b \sinh ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.43, size = 117, normalized size = 0.56 \[ -\frac {\text {csch}^2(e+f x) \left (\left (-8 a^2+40 a b-35 b^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b \sinh ^2(e+f x)}{a}+1\right )+3 a \text {csch}^2(e+f x) \left (2 a \text {csch}^2(e+f x)+8 a-7 b\right )\right )}{24 a^3 f \sqrt {a+b \sinh ^2(e+f x)} \left (a \text {csch}^2(e+f x)+b\right )} \]
Antiderivative was successfully verified.
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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.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 73, normalized size = 0.35 \[ \frac {\mathit {`\,int/indef0`\,}\left (\frac {\cosh ^{4}\left (f x +e \right )}{\left (b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )+2 a b \left (\sinh ^{2}\left (f x +e \right )\right )+a^{2}\right ) \sinh \left (f x +e \right )^{5} \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth \left (f x + e\right )^{5}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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