Optimal. Leaf size=371 \[ -\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{32 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{32 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}-\frac{3 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{3 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}-\frac{c}{4 b d (c \sec (a+b x))^{7/2} \sqrt{d \csc (a+b x)}}+\frac{1}{16 b c d (c \sec (a+b x))^{3/2} \sqrt{d \csc (a+b x)}} \]
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Rubi [A] time = 0.290564, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {2627, 2628, 2629, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{32 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right ) \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)}}{32 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}-\frac{3 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{3 \sqrt{\tan (a+b x)} \sqrt{d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}-\frac{c}{4 b d (c \sec (a+b x))^{7/2} \sqrt{d \csc (a+b x)}}+\frac{1}{16 b c d (c \sec (a+b x))^{3/2} \sqrt{d \csc (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 2628
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} (c \sec (a+b x))^{5/2}} \, dx &=-\frac{c}{4 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac{\int \frac{\sqrt{d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx}{8 d^2}\\ &=-\frac{c}{4 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac{1}{16 b c d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac{3 \int \frac{\sqrt{d \csc (a+b x)}}{\sqrt{c \sec (a+b x)}} \, dx}{32 c^2 d^2}\\ &=-\frac{c}{4 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac{1}{16 b c d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac{\left (3 \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}\right ) \int \frac{1}{\sqrt{\tan (a+b x)}} \, dx}{32 c^2 d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{4 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac{1}{16 b c d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac{\left (3 \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 )}{32 b c^2 d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{4 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac{1}{16 b c d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac{\left (3 \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 )}{16 b c^2 d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{4 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac{1}{16 b c d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac{\left (3 \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 )}{32 b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{\left (3 \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 )}{32 b c^2 d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{4 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac{1}{16 b c d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac{\left (3 \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 )}{64 b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{\left (3 \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 )}{64 b c^2 d^2 \sqrt{c \sec (a+b x)}}-\frac{\left (3 \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 )}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}-\frac{\left (3 \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 )}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{4 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac{1}{16 b c d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac{3 \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)}}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{3 \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)}}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{\left (3 \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 )}{32 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}-\frac{\left (3 \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 )}{32 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}\\ &=-\frac{c}{4 b d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac{1}{16 b c d \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}{32 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{3 \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (a+b x)}\right ) \sqrt{d \csc (a+b x)} \sqrt{\tan (a+b x)}}{32 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}-\frac{3 \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)}}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}+\frac{3 \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)}}{64 \sqrt{2} b c^2 d^2 \sqrt{c \sec (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.378894, size = 81, normalized size = 0.22 \[ -\frac{\csc ^3(a+b x) \left (2 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(a+b x)\right )+\cos (2 (a+b x))-\cos (4 (a+b x))\right )}{32 b c (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.183, size = 696, normalized size = 1.9 \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}} \left (c \sec \left (b x + a\right )\right )^{\frac{5}{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}{\left (d \csc \left (b x + a\right )\right )^{\frac{3}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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