Optimal. Leaf size=89 \[ \frac{2 c d \sqrt{c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac{2 c^2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b \sqrt{\sin (2 a+2 b x)} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.143064, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2626, 2630, 2572, 2639} \[ \frac{2 c d \sqrt{c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac{2 c^2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b \sqrt{\sin (2 a+2 b x)} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2626
Rule 2630
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int \frac{(c \sec (a+b x))^{3/2}}{\sqrt{d \csc (a+b x)}} \, dx &=\frac{2 c d \sqrt{c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\left (2 c^2\right ) \int \frac{1}{\sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)}} \, dx\\ &=\frac{2 c d \sqrt{c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac{\left (2 c^2\right ) \int \sqrt{c \cos (a+b x)} \sqrt{d \sin (a+b x)} \, dx}{\sqrt{c \cos (a+b x)} \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{d \sin (a+b x)}}\\ &=\frac{2 c d \sqrt{c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac{\left (2 c^2\right ) \int \sqrt{\sin (2 a+2 b x)} \, dx}{\sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{\sin (2 a+2 b x)}}\\ &=\frac{2 c d \sqrt{c \sec (a+b x)}}{b (d \csc (a+b x))^{3/2}}-\frac{2 c^2 E\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{b \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [C] time = 0.366759, size = 66, normalized size = 0.74 \[ -\frac{2 c d \sqrt{c \sec (a+b x)} \left (\sqrt [4]{-\cot ^2(a+b x)} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{4},\frac{1}{2},\csc ^2(a+b x)\right )-1\right )}{b (d \csc (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.191, size = 505, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}}{\sqrt{d \csc \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \csc \left (b x + a\right )} \sqrt{c \sec \left (b x + a\right )} c \sec \left (b x + a\right )}{d \csc \left (b x + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}}{\sqrt{d \csc \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]