Optimal. Leaf size=133 \[ \frac {6 c^2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{5 b d^4 \sqrt {\sin (2 a+2 b x)}}-\frac {6 c (c \sin (a+b x))^{3/2}}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 c (c \sin (a+b x))^{3/2}}{5 b d (d \cos (a+b x))^{5/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2566, 2571, 2572, 2639} \[ \frac {6 c^2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{5 b d^4 \sqrt {\sin (2 a+2 b x)}}-\frac {6 c (c \sin (a+b x))^{3/2}}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {2 c (c \sin (a+b x))^{3/2}}{5 b d (d \cos (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2571
Rule 2572
Rule 2639
Rubi steps
\begin {align*} \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{7/2}} \, dx &=\frac {2 c (c \sin (a+b x))^{3/2}}{5 b d (d \cos (a+b x))^{5/2}}-\frac {\left (3 c^2\right ) \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{3/2}} \, dx}{5 d^2}\\ &=\frac {2 c (c \sin (a+b x))^{3/2}}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 c (c \sin (a+b x))^{3/2}}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {\left (6 c^2\right ) \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx}{5 d^4}\\ &=\frac {2 c (c \sin (a+b x))^{3/2}}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 c (c \sin (a+b x))^{3/2}}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {\left (6 c^2 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{5 d^4 \sqrt {\sin (2 a+2 b x)}}\\ &=\frac {2 c (c \sin (a+b x))^{3/2}}{5 b d (d \cos (a+b x))^{5/2}}-\frac {6 c (c \sin (a+b x))^{3/2}}{5 b d^3 \sqrt {d \cos (a+b x)}}+\frac {6 c^2 \sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{5 b d^4 \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 70, normalized size = 0.53 \[ \frac {2 \cos ^2(a+b x)^{5/4} \cot (a+b x) (c \sin (a+b x))^{9/2} \, _2F_1\left (\frac {7}{4},\frac {9}{4};\frac {11}{4};\sin ^2(a+b x)\right )}{7 b c^2 (d \cos (a+b x))^{7/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )}}{d^{4} \cos \left (b x + a\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 531, normalized size = 3.99 \[ -\frac {\left (6 \left (\cos ^{3}\left (b x +a \right )\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 )}}\, \EllipticE \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 )-3 \left (\cos ^{3}\left (b x +a \right )\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 )+6 \left (\cos ^{2}\left (b x +a \right )\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 )}}\, \EllipticE \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 )-3 \left (\cos ^{2}\left (b x +a \right )\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 )-3 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}+4 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-\sqrt {2}\right ) \left (c \sin \left (b x +a \right )\right )^{\frac {5}{2}} \cos \left (b x +a \right ) \sqrt {2}}{5 b \sin \left (b x +a \right )^{3} \left (d \cos \left (b x +a \right )\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^{5/2}}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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