Optimal. Leaf size=315 \[ \frac{c^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right )}{\sqrt{2} b d^{5/2}}-\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}+1\right )}{\sqrt{2} b d^{5/2}}-\frac{c^{5/2} \log \left (-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right )}{2 \sqrt{2} b d^{5/2}}+\frac{c^{5/2} \log \left (\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right )}{2 \sqrt{2} b d^{5/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.2657, antiderivative size = 315, 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, 2574, 297, 1162, 617, 204, 1165, 628} \[ \frac{c^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right )}{\sqrt{2} b d^{5/2}}-\frac{c^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}+1\right )}{\sqrt{2} b d^{5/2}}-\frac{c^{5/2} \log \left (-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right )}{2 \sqrt{2} b d^{5/2}}+\frac{c^{5/2} \log \left (\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right )}{2 \sqrt{2} b d^{5/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2574
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx &=\frac{2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{c^2 \int \frac{\sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}} \, dx}{d^2}\\ &=\frac{2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{\left (2 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{c^2+d^2 x^4} \, dx,x,\frac{\sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}\right )}{b d}\\ &=\frac{2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}+\frac{c^3 \operatorname{Subst}\left (\int \frac{c-d x^2}{c^2+d^2 x^4} \, dx,x,\frac{\sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}\right )}{b d^2}-\frac{c^3 \operatorname{Subst}\left (\int \frac{c+d x^2}{c^2+d^2 x^4} \, dx,x,\frac{\sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}\right )}{b d^2}\\ &=\frac{2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{\frac{c}{d}-\frac{\sqrt{2} \sqrt{c} x}{\sqrt{d}}+x^2} \, dx,x,\frac{\sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}\right )}{2 b d^3}-\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{\frac{c}{d}+\frac{\sqrt{2} \sqrt{c} x}{\sqrt{d}}+x^2} \, dx,x,\frac{\sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}\right )}{2 b d^3}-\frac{c^{5/2} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{c}}{\sqrt{d}}+2 x}{-\frac{c}{d}-\frac{\sqrt{2} \sqrt{c} x}{\sqrt{d}}-x^2} \, dx,x,\frac{\sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}\right )}{2 \sqrt{2} b d^{5/2}}-\frac{c^{5/2} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{c}}{\sqrt{d}}-2 x}{-\frac{c}{d}+\frac{\sqrt{2} \sqrt{c} x}{\sqrt{d}}-x^2} \, dx,x,\frac{\sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}\right )}{2 \sqrt{2} b d^{5/2}}\\ &=-\frac{c^{5/2} \log \left (\sqrt{c}-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)\right )}{2 \sqrt{2} b d^{5/2}}+\frac{c^{5/2} \log \left (\sqrt{c}+\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)\right )}{2 \sqrt{2} b d^{5/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{c^{5/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right )}{\sqrt{2} b d^{5/2}}+\frac{c^{5/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right )}{\sqrt{2} b d^{5/2}}\\ &=\frac{c^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right )}{\sqrt{2} b d^{5/2}}-\frac{c^{5/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right )}{\sqrt{2} b d^{5/2}}-\frac{c^{5/2} \log \left (\sqrt{c}-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)\right )}{2 \sqrt{2} b d^{5/2}}+\frac{c^{5/2} \log \left (\sqrt{c}+\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)\right )}{2 \sqrt{2} b d^{5/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.125588, size = 67, normalized size = 0.21 \[ \frac{2 \cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{7/2} \, _2F_1\left (\frac{7}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right )}{7 b c d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.093, size = 532, normalized size = 1.7 \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{5}{2}}}{\left (d \cos \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{\left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}}}{\left (d \cos \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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