Optimal. Leaf size=327 \[ \frac{c^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{\sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}-\frac{c^2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{\sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}+\frac{c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}-\frac{c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}+\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}} \]
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Rubi [A] time = 0.21939, antiderivative size = 327, 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 = {2624, 2629, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{c^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{\sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}-\frac{c^2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{\sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}+\frac{c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}-\frac{c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}+\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2624
Rule 2629
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(c \sec (a+b x))^{3/2}}{(d \csc (a+b x))^{3/2}} \, dx &=\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}}-\frac{c^2 \int \frac{\sqrt{d \csc (a+b x)}}{\sqrt{c \sec (a+b x)}} \, dx}{d^2}\\ &=\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}}-\frac{\left (c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \int \frac{1}{\sqrt{\tan (a+b x)}} \, dx}{d^2 \sqrt{c \sec (a+b x)}}\\ &=\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}}-\frac{\left (c^2 \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 d^2 \sqrt{c \sec (a+b x)}}\\ &=\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}}-\frac{\left (2 c^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 d^2 \sqrt{c \sec (a+b x)}}\\ &=\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}}-\frac{\left (c^2 \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 d^2 \sqrt{c \sec (a+b x)}}-\frac{\left (c^2 \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 d^2 \sqrt{c \sec (a+b x)}}\\ &=\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}}-\frac{\left (c^2 \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 d^2 \sqrt{c \sec (a+b x)}}-\frac{\left (c^2 \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 d^2 \sqrt{c \sec (a+b x)}}+\frac{\left (c^2 \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 d^2 \sqrt{c \sec (a+b x)}}+\frac{\left (c^2 \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 d^2 \sqrt{c \sec (a+b x)}}\\ &=\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}}+\frac{c^2 \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 d^2 \sqrt{c \sec (a+b x)}}-\frac{c^2 \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 d^2 \sqrt{c \sec (a+b x)}}-\frac{\left (c^2 \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 d^2 \sqrt{c \sec (a+b x)}}+\frac{\left (c^2 \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 d^2 \sqrt{c \sec (a+b x)}}\\ &=\frac{2 c \sqrt{c \sec (a+b x)}}{b d \sqrt{d \csc (a+b x)}}+\frac{c^2 \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 d^2 \sqrt{c \sec (a+b x)}}-\frac{c^2 \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 d^2 \sqrt{c \sec (a+b x)}}+\frac{c^2 \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 d^2 \sqrt{c \sec (a+b x)}}-\frac{c^2 \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 d^2 \sqrt{c \sec (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.242222, size = 64, normalized size = 0.2 \[ \frac{2 c \sqrt{c \sec (a+b x)} \left (\cot ^2(a+b x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(a+b x)\right )+3\right )}{3 b d \sqrt{d \csc (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.171, size = 656, normalized size = 2. \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{\left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}}{\left (d \csc \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{\left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}}{\left (d \csc \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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