Optimal. Leaf size=121 \[ \frac{2 a \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right ) \sqrt{e \csc (c+d x)}}{d}+\frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}+\frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.137354, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3878, 3872, 2838, 2564, 329, 212, 206, 203, 2641} \[ \frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}+\frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}+\frac{2 a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \csc (c+d x)}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3878
Rule 3872
Rule 2838
Rule 2564
Rule 329
Rule 212
Rule 206
Rule 203
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{e \csc (c+d x)} (a+a \sec (c+d x)) \, dx &=\left (\sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{a+a \sec (c+d x)}{\sqrt{\sin (c+d x)}} \, dx\\ &=-\left (\left (\sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{(-a-a \cos (c+d x)) \sec (c+d x)}{\sqrt{\sin (c+d x)}} \, dx\right )\\ &=\left (a \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx+\left (a \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\sec (c+d x)}{\sqrt{\sin (c+d x)}} \, dx\\ &=\frac{2 a \sqrt{e \csc (c+d x)} F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d}+\frac{\left (a \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{2 a \sqrt{e \csc (c+d x)} F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d}+\frac{\left (2 a \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d}\\ &=\frac{2 a \sqrt{e \csc (c+d x)} F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d}+\frac{\left (a \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d}+\frac{\left (a \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d}\\ &=\frac{a \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right ) \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}{d}+\frac{a \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right ) \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}{d}+\frac{2 a \sqrt{e \csc (c+d x)} F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.835445, size = 111, normalized size = 0.92 \[ -\frac{a \sqrt{e \csc (c+d x)} \left (4 \sqrt{\sin (c+d x)} \sqrt{\csc (c+d x)} \text{EllipticF}\left (\frac{1}{4} (-2 c-2 d x+\pi ),2\right )+\log \left (1-\sqrt{\csc (c+d x)}\right )-\log \left (\sqrt{\csc (c+d x)}+1\right )+2 \tan ^{-1}\left (\sqrt{\csc (c+d x)}\right )\right )}{2 d \sqrt{\csc (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.209, size = 288, normalized size = 2.4 \begin{align*} -{\frac{a\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}\sqrt{{\frac{e}{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{-i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) +i}{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }}} \left ( i{\it EllipticPi} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +i{\it EllipticPi} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +{\it EllipticPi} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) \right ) } \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 \sqrt{e \csc \left (d x + c\right )}{\left (a \sec \left (d x + c\right ) + 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 (\sqrt{e \csc \left (d x + c\right )}{\left (a \sec \left (d x + c\right ) + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \sqrt{e \csc{\left (c + d x \right )}}\, dx + \int \sqrt{e \csc{\left (c + d x \right )}} \sec{\left (c + d x \right )}\, dx\right ) \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 \sqrt{e \csc \left (d x + c\right )}{\left (a \sec \left (d x + c\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]