Optimal. Leaf size=110 \[ \frac{10 a^3 \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac{10 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt{e \cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.202305, antiderivative size = 110, 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 = {2670, 2680, 2682, 2642, 2641} \[ \frac{10 a^3 \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac{10 a^3 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d e^2 \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2670
Rule 2680
Rule 2682
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{5/2}} \, dx &=\frac{a^6 \int \frac{(e \cos (c+d x))^{7/2}}{(a-a \sin (c+d x))^3} \, dx}{e^6}\\ &=\frac{4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac{\left (5 a^4\right ) \int \frac{(e \cos (c+d x))^{3/2}}{a-a \sin (c+d x)} \, dx}{3 e^4}\\ &=\frac{10 a^3 \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac{\left (5 a^3\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{3 e^2}\\ &=\frac{10 a^3 \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac{\left (5 a^3 \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{10 a^3 \sqrt{e \cos (c+d x)}}{3 d e^3}-\frac{10 a^3 \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{4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}\\ \end{align*}
Mathematica [C] time = 0.0527323, size = 66, normalized size = 0.6 \[ \frac{8 \sqrt [4]{2} a^3 (\sin (c+d x)+1)^{3/4} \, _2F_1\left (-\frac{5}{4},-\frac{3}{4};\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.744, size = 219, normalized size = 2. \begin{align*}{\frac{2\,{a}^{3}}{3\,d{e}^{2}} \left ( 10\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-12\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-5\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +12\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-7\,\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) ^{-1} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \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 (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{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^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{e \cos \left (d x + c\right )}}{e^{3} \cos \left (d x + c\right )^{3}}, 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 (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]