Optimal. Leaf size=160 \[ \frac{2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac{2 a \left (a^2+6 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.241024, antiderivative size = 160, 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, 2862, 2669, 2640, 2639} \[ \frac{2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac{2 a \left (a^2+6 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2691
Rule 2862
Rule 2669
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx &=\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}}-\frac{2 \int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x)) \left (\frac{a^2}{2}+2 b^2+\frac{5}{2} a b \sin (c+d x)\right ) \, dx}{e^2}\\ &=\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}}-\frac{4 \int \sqrt{e \cos (c+d x)} \left (\frac{5}{4} a \left (a^2+6 b^2\right )+\frac{5}{4} b \left (3 a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx}{5 e^2}\\ &=\frac{2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}}-\frac{\left (a \left (a^2+6 b^2\right )\right ) \int \sqrt{e \cos (c+d x)} \, dx}{e^2}\\ &=\frac{2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}}-\frac{\left (a \left (a^2+6 b^2\right ) \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{e^2 \sqrt{\cos (c+d x)}}\\ &=\frac{2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac{2 a \left (a^2+6 b^2\right ) \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac{2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.396142, size = 98, normalized size = 0.61 \[ \frac{6 \left (a \left (a^2+3 b^2\right ) \sin (c+d x)+3 a^2 b+b^3\right )-6 a \left (a^2+6 b^2\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 b^3 \cos ^2(c+d x)}{3 d e \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.937, size = 248, normalized size = 1.6 \begin{align*} -{\frac{2}{3\,de} \left ( -4\,{b}^{3} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+3\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ){a}^{3}+18\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) a{b}^{2}-6\,{a}^{3}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-18\,a{b}^{2}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}{b}^{3}-9\,{a}^{2}b\sin \left ( 1/2\,dx+c/2 \right ) -4\,{b}^{3}\sin \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2} \cos \left (d x + c\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]