Optimal. Leaf size=313 \[ -\frac{c^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}\right )}{\sqrt{2} b d^{3/2}}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}+1\right )}{\sqrt{2} b d^{3/2}}+\frac{c^{3/2} \log \left (-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right )}{2 \sqrt{2} b d^{3/2}}-\frac{c^{3/2} \log \left (\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right )}{2 \sqrt{2} b d^{3/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{b d \sqrt{d \cos (a+b x)}} \]
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Rubi [A] time = 0.275586, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {2566, 2575, 297, 1162, 617, 204, 1165, 628} \[ -\frac{c^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}\right )}{\sqrt{2} b d^{3/2}}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}+1\right )}{\sqrt{2} b d^{3/2}}+\frac{c^{3/2} \log \left (-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right )}{2 \sqrt{2} b d^{3/2}}-\frac{c^{3/2} \log \left (\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right )}{2 \sqrt{2} b d^{3/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{b d \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2575
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx &=\frac{2 c \sqrt{c \sin (a+b x)}}{b d \sqrt{d \cos (a+b x)}}-\frac{c^2 \int \frac{\sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}} \, dx}{d^2}\\ &=\frac{2 c \sqrt{c \sin (a+b x)}}{b d \sqrt{d \cos (a+b x)}}+\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{d^2+c^2 x^4} \, dx,x,\frac{\sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{b d}\\ &=\frac{2 c \sqrt{c \sin (a+b x)}}{b d \sqrt{d \cos (a+b x)}}-\frac{c^2 \operatorname{Subst}\left (\int \frac{d-c x^2}{d^2+c^2 x^4} \, dx,x,\frac{\sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{b d}+\frac{c^2 \operatorname{Subst}\left (\int \frac{d+c x^2}{d^2+c^2 x^4} \, dx,x,\frac{\sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{b d}\\ &=\frac{2 c \sqrt{c \sin (a+b x)}}{b d \sqrt{d \cos (a+b x)}}+\frac{c^{3/2} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{d}}{\sqrt{c}}+2 x}{-\frac{d}{c}-\frac{\sqrt{2} \sqrt{d} x}{\sqrt{c}}-x^2} \, dx,x,\frac{\sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{2 \sqrt{2} b d^{3/2}}+\frac{c^{3/2} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{d}}{\sqrt{c}}-2 x}{-\frac{d}{c}+\frac{\sqrt{2} \sqrt{d} x}{\sqrt{c}}-x^2} \, dx,x,\frac{\sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{2 \sqrt{2} b d^{3/2}}+\frac{c \operatorname{Subst}\left (\int \frac{1}{\frac{d}{c}-\frac{\sqrt{2} \sqrt{d} x}{\sqrt{c}}+x^2} \, dx,x,\frac{\sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{2 b d}+\frac{c \operatorname{Subst}\left (\int \frac{1}{\frac{d}{c}+\frac{\sqrt{2} \sqrt{d} x}{\sqrt{c}}+x^2} \, dx,x,\frac{\sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{2 b d}\\ &=\frac{c^{3/2} \log \left (\sqrt{d}+\sqrt{d} \cot (a+b x)-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{2 \sqrt{2} b d^{3/2}}-\frac{c^{3/2} \log \left (\sqrt{d}+\sqrt{d} \cot (a+b x)+\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{2 \sqrt{2} b d^{3/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{b d \sqrt{d \cos (a+b x)}}+\frac{c^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}\right )}{\sqrt{2} b d^{3/2}}-\frac{c^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}\right )}{\sqrt{2} b d^{3/2}}\\ &=-\frac{c^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}\right )}{\sqrt{2} b d^{3/2}}+\frac{c^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}\right )}{\sqrt{2} b d^{3/2}}+\frac{c^{3/2} \log \left (\sqrt{d}+\sqrt{d} \cot (a+b x)-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{2 \sqrt{2} b d^{3/2}}-\frac{c^{3/2} \log \left (\sqrt{d}+\sqrt{d} \cot (a+b x)+\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}\right )}{2 \sqrt{2} b d^{3/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{b d \sqrt{d \cos (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.163983, size = 67, normalized size = 0.21 \[ \frac{2 \sqrt [4]{\cos ^2(a+b x)} (c \sin (a+b x))^{5/2} \, _2F_1\left (\frac{5}{4},\frac{5}{4};\frac{9}{4};\sin ^2(a+b x)\right )}{5 b c d \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.092, size = 642, 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{\left (c \sin \left (b x + a\right )\right )^{\frac{3}{2}}}{\left (d \cos \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 \sin \left (b x + a\right )\right )^{\frac{3}{2}}}{\left (d \cos \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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