Optimal. Leaf size=165 \[ -\frac {2 \left (3 a^2-2 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d e^4 \sqrt {\sin (c+d x)}}-\frac {2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 a b}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2691, 2669, 2636, 2640, 2639} \[ -\frac {2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 \left (3 a^2-2 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d e^4 \sqrt {\sin (c+d x)}}-\frac {2 a b}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2640
Rule 2669
Rule 2691
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{7/2}} \, dx &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac {2 \int \frac {-\frac {3 a^2}{2}+b^2-\frac {1}{2} a b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac {2 a b}{5 d e^3 \sqrt {e \sin (c+d x)}}+\frac {\left (3 a^2-2 b^2\right ) \int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac {2 a b}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {\left (3 a^2-2 b^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{5 e^4}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac {2 a b}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {\left (\left (3 a^2-2 b^2\right ) \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 e^4 \sqrt {\sin (c+d x)}}\\ &=-\frac {2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac {2 a b}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {2 \left (3 a^2-2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d e^4 \sqrt {\sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 109, normalized size = 0.66 \[ -\frac {\left (7 a^2+2 b^2\right ) \cos (c+d x)-4 \left (3 a^2-2 b^2\right ) \sin ^{\frac {5}{2}}(c+d x) E\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )-3 a^2 \cos (3 (c+d x))+8 a b+2 b^2 \cos (3 (c+d x))}{10 d e (e \sin (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {e \sin \left (d x + c\right )}}{e^{4} \cos \left (d x + c\right )^{4} - 2 \, e^{4} \cos \left (d x + c\right )^{2} + e^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 327, normalized size = 1.98 \[ \frac {-\frac {4 a b}{5 e \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {\left (6 a^{2}-4 b^{2}\right ) \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+\left (-8 a^{2}+2 b^{2}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}-4 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}-3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}}{5 e^{3} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d} \]
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 \[ \int \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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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