Optimal. Leaf size=114 \[ \frac {2 \left (3 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d \sqrt {e \sin (c+d x)}}+\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}{3 d e} \]
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Rubi [A] time = 0.13, antiderivative size = 114, 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 = {2692, 2669, 2642, 2641} \[ \frac {2 \left (3 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d \sqrt {e \sin (c+d x)}}+\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}{3 d e} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 2669
Rule 2692
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^2}{\sqrt {e \sin (c+d x)}} \, dx &=\frac {2 b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2}{3} \int \frac {\frac {3 a^2}{2}+b^2+\frac {5}{2} a b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e}+\frac {1}{3} \left (3 a^2+2 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e}+\frac {\left (\left (3 a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 \sqrt {e \sin (c+d x)}}\\ &=\frac {2 \left (3 a^2+2 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}+\frac {10 a b \sqrt {e \sin (c+d x)}}{3 d e}+\frac {2 b (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{3 d e}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 79, normalized size = 0.69 \[ \frac {2 b \sin (c+d x) (6 a+b \cos (c+d x))-2 \left (3 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )}{3 d \sqrt {e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.17, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {e \sin \left (d x + c\right )}}{e \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \sin \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 170, normalized size = 1.49 \[ -\frac {3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}-2 b^{2} \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )-12 a b \sin \left (d x +c \right ) \cos \left (d x +c \right )}{3 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \sin \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2}{\sqrt {e\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\left (a + b \cos {\left (c + d x \right )}\right )^{2}}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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