Optimal. Leaf size=270 \[ -\frac{\sqrt{\tan (a+b x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{d \csc (a+b x)}}{\sqrt{2} b \sqrt{c \sec (a+b x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{\sqrt{2} b \sqrt{c \sec (a+b x)}}-\frac{\sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}}+\frac{\sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}} \]
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Rubi [A] time = 0.130926, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {2629, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt{\tan (a+b x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{d \csc (a+b x)}}{\sqrt{2} b \sqrt{c \sec (a+b x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{\sqrt{2} b \sqrt{c \sec (a+b x)}}-\frac{\sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}}+\frac{\sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2629
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{d \csc (a+b x)}}{\sqrt{c \sec (a+b x)}} \, dx &=\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \int \frac{1}{\sqrt{\tan (a+b x)}} \, dx}{\sqrt{c \sec (a+b x)}}\\ &=\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\tan (a+b x)\right )}{b \sqrt{c \sec (a+b x)}}\\ &=\frac{\left (2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{\tan (a+b x)}\right )}{b \sqrt{c \sec (a+b x)}}\\ &=\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (a+b x)}\right )}{b \sqrt{c \sec (a+b x)}}+\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (a+b x)}\right )}{b \sqrt{c \sec (a+b x)}}\\ &=\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (a+b x)}\right )}{2 b \sqrt{c \sec (a+b x)}}+\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (a+b x)}\right )}{2 b \sqrt{c \sec (a+b x)}}-\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (a+b x)}\right )}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}}-\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (a+b x)}\right )}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}}\\ &=-\frac{\sqrt{d \csc (a+b x)} \log \left (1-\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{\tan (a+b x)}}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}}+\frac{\sqrt{d \csc (a+b x)} \log \left (1+\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{\tan (a+b x)}}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}}+\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (a+b x)}\right )}{\sqrt{2} b \sqrt{c \sec (a+b x)}}-\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (a+b x)}\right )}{\sqrt{2} b \sqrt{c \sec (a+b x)}}\\ &=-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}{\sqrt{2} b \sqrt{c \sec (a+b x)}}+\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}{\sqrt{2} b \sqrt{c \sec (a+b x)}}-\frac{\sqrt{d \csc (a+b x)} \log \left (1-\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{\tan (a+b x)}}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}}+\frac{\sqrt{d \csc (a+b x)} \log \left (1+\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{\tan (a+b x)}}{2 \sqrt{2} b \sqrt{c \sec (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.125277, size = 55, normalized size = 0.2 \[ -\frac{2 \cot (a+b x) \sqrt{d \csc (a+b x)} \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(a+b x)\right )}{3 b \sqrt{c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.184, size = 322, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{2\,b\cos \left ( bx+a \right ) \left ( -1+\cos \left ( bx+a \right ) \right ) }\sqrt{{\frac{d}{\sin \left ( bx+a \right ) }}}\sqrt{-{\frac{-1+\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}} \left ( i{\it EllipticPi} \left ( \sqrt{-{\frac{-1+\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -i{\it EllipticPi} \left ( \sqrt{-{\frac{-1+\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -2\,{\it EllipticF} \left ( \sqrt{-{\frac{-1+\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},1/2\,\sqrt{2} \right ) +{\it EllipticPi} \left ( \sqrt{-{\frac{-1+\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) +{\it EllipticPi} \left ( \sqrt{-{\frac{-1+\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{{\frac{c}{\cos \left ( bx+a \right ) }}}}}} \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 \frac{\sqrt{d \csc \left (b x + a\right )}}{\sqrt{c \sec \left (b x + a\right )}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \csc{\left (a + b x \right )}}}{\sqrt{c \sec{\left (a + b x \right )}}}\, dx \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 \frac{\sqrt{d \csc \left (b x + a\right )}}{\sqrt{c \sec \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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