Optimal. Leaf size=187 \[ \frac{2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac{2 \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a \left (a^2-2 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.264079, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2691, 2861, 2669, 2640, 2639} \[ \frac{2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac{2 \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a \left (a^2-2 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2691
Rule 2861
Rule 2669
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx &=\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac{2 \int \frac{(a+b \sin (c+d x)) \left (-\frac{3 a^2}{2}+2 b^2+\frac{1}{2} a b \sin (c+d x)\right )}{(e \cos (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac{2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{4 \int \sqrt{e \cos (c+d x)} \left (-\frac{3}{4} a \left (a^2-2 b^2\right )-\frac{3}{4} b \left (3 a^2-4 b^2\right ) \sin (c+d x)\right ) \, dx}{5 e^4}\\ &=\frac{2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac{2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{\left (3 a \left (a^2-2 b^2\right )\right ) \int \sqrt{e \cos (c+d x)} \, dx}{5 e^4}\\ &=\frac{2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac{2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{\left (3 a \left (a^2-2 b^2\right ) \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 e^4 \sqrt{\cos (c+d x)}}\\ &=\frac{2 b \left (3 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac{6 a \left (a^2-2 b^2\right ) \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}-\frac{2 (a+b \sin (c+d x)) \left (a b-\left (3 a^2-4 b^2\right ) \sin (c+d x)\right )}{5 d e^3 \sqrt{e \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.704941, size = 126, normalized size = 0.67 \[ \frac{2 \left (b \left (3 a^2+b^2\right ) \sec ^2(c+d x)-3 a \left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+a \left (a^2+3 b^2\right ) \tan (c+d x) \sec (c+d x)+3 a^3 \sin (c+d x)-6 a b^2 \sin (c+d x)-5 b^3\right )}{5 d e^3 \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.974, size = 618, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{4} \cos \left (d x + c\right )^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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