Optimal. Leaf size=327 \[ \frac{d^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{c \sec (a+b x)}}{\sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{d^2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{c \sec (a+b x)}}{\sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{d^2 \sqrt{c \sec (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{d^2 \sqrt{c \sec (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}} \]
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Rubi [A] time = 0.215005, 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 = {2623, 2629, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{d^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{c \sec (a+b x)}}{\sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{d^2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{c \sec (a+b x)}}{\sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{d^2 \sqrt{c \sec (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{d^2 \sqrt{c \sec (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{2 \sqrt{2} b c^2 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2623
Rule 2629
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(d \csc (a+b x))^{3/2}}{(c \sec (a+b x))^{3/2}} \, dx &=-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}}-\frac{d^2 \int \frac{\sqrt{c \sec (a+b x)}}{\sqrt{d \csc (a+b x)}} \, dx}{c^2}\\ &=-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}}-\frac{\left (d^2 \sqrt{c \sec (a+b x)}\right ) \int \sqrt{\tan (a+b x)} \, dx}{c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}}-\frac{\left (d^2 \sqrt{c \sec (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1+x^2} \, dx,x,\tan (a+b x)\right )}{b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}}-\frac{\left (2 d^2 \sqrt{c \sec (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{\tan (a+b x)}\right )}{b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}}+\frac{\left (d^2 \sqrt{c \sec (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (a+b x)}\right )}{b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{\left (d^2 \sqrt{c \sec (a+b x)}\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (a+b x)}\right )}{b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}}-\frac{\left (d^2 \sqrt{c \sec (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 c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{\left (d^2 \sqrt{c \sec (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 c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{\left (d^2 \sqrt{c \sec (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 c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{\left (d^2 \sqrt{c \sec (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 c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}}-\frac{d^2 \log \left (1-\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{c \sec (a+b x)}}{2 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{d^2 \log \left (1+\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{c \sec (a+b x)}}{2 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{\left (d^2 \sqrt{c \sec (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 c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{\left (d^2 \sqrt{c \sec (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 c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ &=-\frac{2 d \sqrt{d \csc (a+b x)}}{b c \sqrt{c \sec (a+b x)}}+\frac{d^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{c \sec (a+b x)}}{\sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{d^2 \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{c \sec (a+b x)}}{\sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}-\frac{d^2 \log \left (1-\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{c \sec (a+b x)}}{2 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}+\frac{d^2 \log \left (1+\sqrt{2} \sqrt{\tan (a+b x)}+\tan (a+b x)\right ) \sqrt{c \sec (a+b x)}}{2 \sqrt{2} b c^2 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.634882, size = 199, normalized size = 0.61 \[ -\frac{d \sqrt{d \csc (a+b x)} \left (8 \sqrt [4]{\cot ^2(a+b x)}+\sqrt{2} \log \left (\sqrt{\cot ^2(a+b x)}-\sqrt{2} \sqrt [4]{\cot ^2(a+b x)}+1\right )-\sqrt{2} \log \left (\sqrt{\cot ^2(a+b x)}+\sqrt{2} \sqrt [4]{\cot ^2(a+b x)}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{\cot ^2(a+b x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt [4]{\cot ^2(a+b x)}+1\right )\right )}{4 b c \sqrt [4]{\cot ^2(a+b x)} \sqrt{c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.178, size = 975, normalized size = 3. \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 (d \csc \left (b x + a\right )\right )^{\frac{3}{2}}}{\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{\left (d \csc \left (b x + a\right )\right )^{\frac{3}{2}}}{\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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