Optimal. Leaf size=153 \[ -\frac{2 (e \cos (c+d x))^{3/2}}{15 d e \left (a^3 \sin (c+d x)+a^3\right )}-\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{15 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{15 a d e (a \sin (c+d x)+a)^2}-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a \sin (c+d x)+a)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.173375, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2681, 2683, 2640, 2639} \[ -\frac{2 (e \cos (c+d x))^{3/2}}{15 d e \left (a^3 \sin (c+d x)+a^3\right )}-\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{15 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{15 a d e (a \sin (c+d x)+a)^2}-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2681
Rule 2683
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^3} \, dx &=-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a+a \sin (c+d x))^3}+\frac{\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^2} \, dx}{3 a}\\ &=-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a+a \sin (c+d x))^3}-\frac{2 (e \cos (c+d x))^{3/2}}{15 a d e (a+a \sin (c+d x))^2}+\frac{\int \frac{\sqrt{e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx}{15 a^2}\\ &=-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a+a \sin (c+d x))^3}-\frac{2 (e \cos (c+d x))^{3/2}}{15 a d e (a+a \sin (c+d x))^2}-\frac{2 (e \cos (c+d x))^{3/2}}{15 d e \left (a^3+a^3 \sin (c+d x)\right )}-\frac{\int \sqrt{e \cos (c+d x)} \, dx}{15 a^3}\\ &=-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a+a \sin (c+d x))^3}-\frac{2 (e \cos (c+d x))^{3/2}}{15 a d e (a+a \sin (c+d x))^2}-\frac{2 (e \cos (c+d x))^{3/2}}{15 d e \left (a^3+a^3 \sin (c+d x)\right )}-\frac{\sqrt{e \cos (c+d x)} \int \sqrt{\cos (c+d x)} \, dx}{15 a^3 \sqrt{\cos (c+d x)}}\\ &=-\frac{2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a+a \sin (c+d x))^3}-\frac{2 (e \cos (c+d x))^{3/2}}{15 a d e (a+a \sin (c+d x))^2}-\frac{2 (e \cos (c+d x))^{3/2}}{15 d e \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.0426006, size = 66, normalized size = 0.43 \[ -\frac{(e \cos (c+d x))^{3/2} \, _2F_1\left (\frac{3}{4},\frac{13}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{6 \sqrt [4]{2} a^3 d e (\sin (c+d x)+1)^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 2.585, size = 512, normalized size = 3.4 \begin{align*} \text{result too large to display} \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{\sqrt{e \cos \left (d x + c\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{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{\sqrt{e \cos \left (d x + c\right )}}{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 )}, 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{\sqrt{e \cos \left (d x + c\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]