Optimal. Leaf size=322 \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{4 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{4 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}-\frac{\sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{8 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}+\frac{\sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{8 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}-\frac{c}{2 b d (c \sec (a+b x))^{3/2} \sqrt{d \csc (a+b x)}} \]
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Rubi [A] time = 0.204094, 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 = {2627, 2629, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{4 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{4 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}-\frac{\sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{8 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}+\frac{\sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{8 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}-\frac{c}{2 b d (c \sec (a+b x))^{3/2} \sqrt{d \csc (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 2629
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}} \, dx &=-\frac{c}{2 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac{\int \frac{\sqrt{d \csc (a+b x)}}{\sqrt{c \sec (a+b x)}} \, dx}{4 d^2}\\ &=-\frac{c}{2 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac{\left (\sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \int \frac{1}{\sqrt{\tan (a+b x)}} \, dx}{4 d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{2 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\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 )}{4 b d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{2 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac{\left (\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 )}{2 b d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{2 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\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 )}{4 b d^2 \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 )}{4 b d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{2 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\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 )}{8 b d^2 \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 )}{8 b d^2 \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 )}{8 \sqrt{2} b d^2 \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 )}{8 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{2 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\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)}}{8 \sqrt{2} b d^2 \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)}}{8 \sqrt{2} b d^2 \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 )}{4 \sqrt{2} b d^2 \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 )}{4 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{2 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}{4 \sqrt{2} b d^2 \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)}}{4 \sqrt{2} b d^2 \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)}}{8 \sqrt{2} b d^2 \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)}}{8 \sqrt{2} b d^2 \sqrt{c \sec (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.255619, size = 66, normalized size = 0.2 \[ -\frac{\cot (a+b x) \left (\csc ^2(a+b x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(a+b x)\right )+3\right )}{6 b \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.17, size = 668, normalized size = 2.1 \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}{\left (d \csc \left (b x + a\right )\right )^{\frac{3}{2}} \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(-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}{\left (d \csc \left (b x + a\right )\right )^{\frac{3}{2}} \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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