Optimal. Leaf size=232 \[ \frac{8 a^2+24 a b+3 b^2}{8 f (a-b)^4 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{8 a^2+24 a b+3 b^2}{24 f (a-b)^3 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{\left (8 a^2+24 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right )}{8 f (a-b)^{9/2}}-\frac{\text{sech}^4(e+f x)}{4 f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(8 a-b) \text{sech}^2(e+f x)}{8 f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.315846, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3194, 89, 78, 51, 63, 208} \[ \frac{8 a^2+24 a b+3 b^2}{8 f (a-b)^4 \sqrt{a+b \sinh ^2(e+f x)}}+\frac{8 a^2+24 a b+3 b^2}{24 f (a-b)^3 \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{\left (8 a^2+24 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right )}{8 f (a-b)^{9/2}}-\frac{\text{sech}^4(e+f x)}{4 f (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(8 a-b) \text{sech}^2(e+f x)}{8 f (a-b)^2 \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 89
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tanh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(1+x)^3 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{2 f}\\ &=-\frac{\text{sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (-4 a-3 b)+2 (a-b) x}{(1+x)^2 (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{4 (a-b) f}\\ &=\frac{(8 a-b) \text{sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{\text{sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{\left (8 a^2+24 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{(1+x) (a+b x)^{5/2}} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b)^2 f}\\ &=\frac{8 a^2+24 a b+3 b^2}{24 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(8 a-b) \text{sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{\text{sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{\left (8 a^2+24 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{(1+x) (a+b x)^{3/2}} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b)^3 f}\\ &=\frac{8 a^2+24 a b+3 b^2}{24 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(8 a-b) \text{sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{\text{sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{8 a^2+24 a b+3 b^2}{8 (a-b)^4 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\left (8 a^2+24 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x}} \, dx,x,\sinh ^2(e+f x)\right )}{16 (a-b)^4 f}\\ &=\frac{8 a^2+24 a b+3 b^2}{24 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(8 a-b) \text{sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{\text{sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{8 a^2+24 a b+3 b^2}{8 (a-b)^4 f \sqrt{a+b \sinh ^2(e+f x)}}+\frac{\left (8 a^2+24 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^2(e+f x)}\right )}{8 (a-b)^4 b f}\\ &=-\frac{\left (8 a^2+24 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sinh ^2(e+f x)}}{\sqrt{a-b}}\right )}{8 (a-b)^{9/2} f}+\frac{8 a^2+24 a b+3 b^2}{24 (a-b)^3 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{(8 a-b) \text{sech}^2(e+f x)}{8 (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac{\text{sech}^4(e+f x)}{4 (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac{8 a^2+24 a b+3 b^2}{8 (a-b)^4 f \sqrt{a+b \sinh ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.58949, size = 114, normalized size = 0.49 \[ \frac{2 \left (8 a^2+24 a b+3 b^2\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b \sinh ^2(e+f x)+a}{a-b}\right )+3 (a-b) \text{sech}^4(e+f x) ((8 a-b) \cosh (2 (e+f x))+4 a+3 b)}{48 f (a-b)^3 \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.298, size = 213, normalized size = 0.9 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ( -{\frac{ \left ( \sinh \left ( fx+e \right ) \right ) ^{5} \left ({b}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{4}+2\,ab \left ( \sinh \left ( fx+e \right ) \right ) ^{2}+{a}^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{4}}{-{b}^{4} \left ( \cosh \left ( fx+e \right ) \right ) ^{18}+ \left ( -4\,a{b}^{3}+4\,{b}^{4} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{16}+ \left ( -6\,{a}^{2}{b}^{2}+12\,a{b}^{3}-6\,{b}^{4} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{14}+ \left ( -4\,{a}^{3}b+12\,{a}^{2}{b}^{2}-12\,a{b}^{3}+4\,{b}^{4} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{12}+ \left ( -{a}^{4}+4\,{a}^{3}b-6\,{a}^{2}{b}^{2}+4\,a{b}^{3}-{b}^{4} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{10}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \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{\tanh \left (f x + e\right )^{5}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (f x + e\right )^{5}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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