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 = 0.28, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, 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 204
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2566
Rule 2575
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*}
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Mathematica [C] time = 0.15, 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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fricas [B] time = 27.41, size = 1865, normalized size = 5.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.13, size = 642, normalized size = 2.05 \[ \frac {\left (i \sin \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \sin \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\sin \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\sin \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+2 \cos \left (b x +a \right ) \sqrt {2}-2 \sqrt {2}\right ) \left (c \sin \left (b x +a \right )\right )^{\frac {3}{2}} \cos \left (b x +a \right ) \sqrt {2}}{2 b \left (-1+\cos \left (b x +a \right )\right ) \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \sin \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}}}{\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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