Optimal. Leaf size=170 \[ \frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-b \tanh ^2(c+d x)+b}}\right )}{d}+\frac {\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a-b \tanh ^2(c+d x)+b}}\right )}{8 d}+\frac {b \tanh (c+d x) \left (a-b \tanh ^2(c+d x)+b\right )^{3/2}}{4 d}+\frac {b (7 a+3 b) \tanh (c+d x) \sqrt {a-b \tanh ^2(c+d x)+b}}{8 d} \]
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Rubi [A] time = 0.19, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 203
Rule 206
Rule 217
Rule 377
Rule 416
Rule 523
Rule 528
Rule 4128
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*}
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Mathematica [A] time = 9.28, size = 280, normalized size = 1.65 \[ \frac {\cosh ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^{5/2} \left (8 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a \sinh ^2(c+d x)+a+b}}\right )+\sqrt {b} \left (15 a^2+10 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a \sinh ^2(c+d x)+a+b}}\right )\right )}{\sqrt {2} d (a \cosh (2 c+2 d x)+a+2 b)^{5/2}}+\frac {\cosh ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^{5/2} \left (\frac {3 \text {sech}(c) \text {sech}^2(c+d x) \left (3 a b \sinh (d x)+b^2 \sinh (d x)\right )}{2 d}+\frac {3 b (3 a+b) \tanh (c) \text {sech}(c+d x)}{2 d}+\frac {b^2 \text {sech}(c) \sinh (d x) \text {sech}^4(c+d x)}{d}+\frac {b^2 \tanh (c) \text {sech}^3(c+d x)}{d}\right )}{(a \cosh (2 c+2 d x)+a+2 b)^2} \]
Antiderivative was successfully verified.
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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 [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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