Optimal. Leaf size=322 \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{c \sec (a+b x)}}{4 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{c \sec (a+b x)}}{4 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{8 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{\sqrt{c \sec (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{8 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{d}{2 b c \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{3/2}} \]
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Rubi [A] time = 0.213415, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2628, 2629, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{c \sec (a+b x)}}{4 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{c \sec (a+b x)}}{4 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{8 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{\sqrt{c \sec (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{8 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{d}{2 b c \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2628
Rule 2629
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx &=\frac{d}{2 b c (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}+\frac{\int \frac{\sqrt{c \sec (a+b x)}}{\sqrt{d \csc (a+b x)}} \, dx}{4 c^2}\\ &=\frac{d}{2 b c (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \int \sqrt{\tan (a+b x)} \, dx}{4 c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=\frac{d}{2 b c (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1+x^2} \, dx,x,\tan (a+b x)\right )}{4 b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=\frac{d}{2 b c (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{\tan (a+b x)}\right )}{2 b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=\frac{d}{2 b c (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}-\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (a+b x)}\right )}{4 b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (a+b x)}\right )}{4 b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=\frac{d}{2 b c (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (a+b x)}\right )}{8 b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (a+b x)}\right )}{8 b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (a+b x)}\right )}{8 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (a+b x)}\right )}{8 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=\frac{d}{2 b c (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}+\frac{\log \left (1-\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{c \sec (a+b x)}}{8 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{\log \left (1+\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{c \sec (a+b x)}}{8 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (a+b x)}\right )}{4 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{\sqrt{c \sec (a+b x)} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (a+b x)}\right )}{4 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=\frac{d}{2 b c (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{c \sec (a+b x)}}{4 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{c \sec (a+b x)}}{4 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{\log \left (1-\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{c \sec (a+b x)}}{8 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{\log \left (1+\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{c \sec (a+b x)}}{8 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ \end{align*}
Mathematica [A] time = 1.97279, size = 223, normalized size = 0.69 \[ \frac{\sqrt{d \csc (a+b x)} \left (4 \sqrt [4]{\cot ^2(a+b x)}-4 \cos (2 (a+b x)) \sqrt [4]{\cot ^2(a+b x)}+\sqrt{2} \log \left (\sqrt{\cot ^2(a+b x)}-\sqrt{2} \sqrt [4]{\cot ^2(a+b x)}+1\right )-\sqrt{2} \log \left (\sqrt{\cot ^2(a+b x)}+\sqrt{2} \sqrt [4]{\cot ^2(a+b x)}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{\cot ^2(a+b x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt [4]{\cot ^2(a+b x)}+1\right )\right )}{16 b c d \sqrt [4]{\cot ^2(a+b x)} \sqrt{c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.19, size = 526, normalized size = 1.6 \begin{align*} \text{result too large to display} \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{1}{\sqrt{d \csc \left (b x + a\right )} \left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}}\,{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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \csc \left (b x + a\right )} \left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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