Optimal. Leaf size=188 \[ \frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 a \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt {e \cos (c+d x)}}+\frac {2 \left (\left (5 a^2-4 b^2\right ) \sin (c+d x)+a b\right ) (a+b \sin (c+d x))}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}} \]
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Rubi [A] time = 0.27, antiderivative size = 188, 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, 2861, 2669, 2642, 2641} \[ \frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 \left (\left (5 a^2-4 b^2\right ) \sin (c+d x)+a b\right ) (a+b \sin (c+d x))}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac {2 a \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt {e \cos (c+d x)}}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 2669
Rule 2691
Rule 2861
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{9/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}-\frac {2 \int \frac {(a+b \sin (c+d x)) \left (-\frac {5 a^2}{2}+2 b^2-\frac {1}{2} a b \sin (c+d x)\right )}{(e \cos (c+d x))^{5/2}} \, dx}{7 e^2}\\ &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac {2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac {4 \int \frac {\frac {1}{4} a \left (5 a^2-6 b^2\right )-\frac {1}{4} b \left (5 a^2-4 b^2\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx}{21 e^4}\\ &=\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac {2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac {\left (a \left (5 a^2-6 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{21 e^4}\\ &=\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac {2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac {\left (a \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 e^4 \sqrt {e \cos (c+d x)}}\\ &=\frac {2 b \left (5 a^2-4 b^2\right ) \sqrt {e \cos (c+d x)}}{21 d e^5}+\frac {2 a \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d e^4 \sqrt {e \cos (c+d x)}}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}+\frac {2 (a+b \sin (c+d x)) \left (a b+\left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{21 d e^3 (e \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 140, normalized size = 0.74 \[ \frac {\sec ^4(c+d x) \sqrt {e \cos (c+d x)} \left (17 a^3 \sin (c+d x)+5 a^3 \sin (3 (c+d x))+4 a \left (5 a^2-6 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+36 a^2 b+30 a b^2 \sin (c+d x)-6 a b^2 \sin (3 (c+d x))-14 b^3 \cos (2 (c+d x))-2 b^3\right )}{42 d e^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{5} \cos \left (d x + c\right )^{5}}, x\right ) \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.73, size = 750, normalized size = 3.99 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \]
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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