Optimal. Leaf size=119 \[ \frac {2 \left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 119, 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 = {2691, 2669, 2642, 2641} \[ \frac {2 \left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 2669
Rule 2691
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}-\frac {2 \int \frac {-\frac {a^2}{2}+b^2+\frac {1}{2} a b \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}+\frac {\left (a^2-2 b^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}+\frac {\left (\left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 e^2 \sqrt {e \cos (c+d x)}}\\ &=\frac {2 a b \sqrt {e \cos (c+d x)}}{3 d e^3}+\frac {2 \left (a^2-2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 72, normalized size = 0.61 \[ \frac {2 \left (\left (a^2+b^2\right ) \sin (c+d x)+\left (a^2-2 b^2\right ) \cos ^{\frac {3}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 a b\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.18, size = 333, normalized size = 2.80 \[ -\frac {2 \left (2 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, a^{2} \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, b^{2} \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}+2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}+2 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d} \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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