Optimal. Leaf size=76 \[ \frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}-\frac {2 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{3 b^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3769, 3771, 2641} \[ \frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}-\frac {2 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{3 b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2641
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{(b \text {sech}(c+d x))^{3/2}} \, dx &=\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}+\frac {\int \sqrt {b \text {sech}(c+d x)} \, dx}{3 b^2}\\ &=\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}+\frac {\left (\sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}\right ) \int \frac {1}{\sqrt {\cosh (c+d x)}} \, dx}{3 b^2}\\ &=-\frac {2 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right ) \sqrt {b \text {sech}(c+d x)}}{3 b^2 d}+\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 63, normalized size = 0.83 \[ \frac {\text {sech}^2(c+d x) \left (\sinh (2 (c+d x))-2 i \sqrt {\cosh (c+d x)} F\left (\left .\frac {1}{2} i (c+d x)\right |2\right )\right )}{3 d (b \text {sech}(c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \operatorname {sech}\left (d x + c\right )}}{b^{2} \operatorname {sech}\left (d x + c\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \operatorname {sech}\left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________