Optimal. Leaf size=170 \[ \frac{\sqrt{b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a-b \tanh ^2(c+d x)+b}}\right )}{8 d}+\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a-b \tanh ^2(c+d x)+b}}\right )}{d}+\frac{b \tanh (c+d x) \left (a-b \tanh ^2(c+d x)+b\right )^{3/2}}{4 d}+\frac{b (7 a+3 b) \tanh (c+d x) \sqrt{a-b \tanh ^2(c+d x)+b}}{8 d} \]
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Rubi [A] time = 0.194354, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4128, 416, 528, 523, 217, 203, 377, 206} \[ \frac{\sqrt{b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a-b \tanh ^2(c+d x)+b}}\right )}{8 d}+\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a-b \tanh ^2(c+d x)+b}}\right )}{d}+\frac{b \tanh (c+d x) \left (a-b \tanh ^2(c+d x)+b\right )^{3/2}}{4 d}+\frac{b (7 a+3 b) \tanh (c+d x) \sqrt{a-b \tanh ^2(c+d x)+b}}{8 d} \]
Antiderivative was successfully verified.
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Rule 4128
Rule 416
Rule 528
Rule 523
Rule 217
Rule 203
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \left (a+b \text{sech}^2(c+d x)\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right )^{5/2}}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b-b x^2} \left ((a+b) (b-4 (a+b))+b (7 a+3 b) x^2\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac{b (7 a+3 b) \tanh (c+d x) \sqrt{a+b-b \tanh ^2(c+d x)}}{8 d}+\frac{b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}+\frac{\operatorname{Subst}\left (\int \frac{(a+b) \left (8 a^2+7 a b+3 b^2\right )-b \left (15 a^2+10 a b+3 b^2\right ) x^2}{\left (1-x^2\right ) \sqrt{a+b-b x^2}} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{b (7 a+3 b) \tanh (c+d x) \sqrt{a+b-b \tanh ^2(c+d x)}}{8 d}+\frac{b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b-b x^2}} \, dx,x,\tanh (c+d x)\right )}{d}+\frac{\left (b \left (15 a^2+10 a b+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b-b x^2}} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{b (7 a+3 b) \tanh (c+d x) \sqrt{a+b-b \tanh ^2(c+d x)}}{8 d}+\frac{b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\tanh (c+d x)}{\sqrt{a+b-b \tanh ^2(c+d x)}}\right )}{d}+\frac{\left (b \left (15 a^2+10 a b+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{\tanh (c+d x)}{\sqrt{a+b-b \tanh ^2(c+d x)}}\right )}{8 d}\\ &=\frac{\sqrt{b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a+b-b \tanh ^2(c+d x)}}\right )}{8 d}+\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a+b-b \tanh ^2(c+d x)}}\right )}{d}+\frac{b (7 a+3 b) \tanh (c+d x) \sqrt{a+b-b \tanh ^2(c+d x)}}{8 d}+\frac{b \tanh (c+d x) \left (a+b-b \tanh ^2(c+d x)\right )^{3/2}}{4 d}\\ \end{align*}
Mathematica [A] time = 9.0174, size = 280, normalized size = 1.65 \[ \frac{\cosh ^5(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^{5/2} \left (\sqrt{b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a \sinh ^2(c+d x)+a+b}}\right )+8 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a \sinh ^2(c+d x)+a+b}}\right )\right )}{\sqrt{2} d (a \cosh (2 c+2 d x)+a+2 b)^{5/2}}+\frac{\cosh ^5(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^{5/2} \left (\frac{3 \text{sech}(c) \text{sech}^2(c+d x) \left (3 a b \sinh (d x)+b^2 \sinh (d x)\right )}{2 d}+\frac{3 b (3 a+b) \tanh (c) \text{sech}(c+d x)}{2 d}+\frac{b^2 \text{sech}(c) \sinh (d x) \text{sech}^4(c+d x)}{d}+\frac{b^2 \tanh (c) \text{sech}^3(c+d x)}{d}\right )}{(a \cosh (2 c+2 d x)+a+2 b)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.193, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) ^{{\frac{5}{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{\left (b \operatorname{sech}\left (d x + c\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{\left (b \operatorname{sech}\left (d x + c\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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