Optimal. Leaf size=63 \[ \frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2}}{3 d e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0508737, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2669, 2640, 2639} \[ \frac{2 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2}}{3 d e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2669
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x)) \, dx &=-\frac{2 b (e \cos (c+d x))^{3/2}}{3 d e}+a \int \sqrt{e \cos (c+d x)} \, dx\\ &=-\frac{2 b (e \cos (c+d x))^{3/2}}{3 d e}+\frac{\left (a \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{2 b (e \cos (c+d x))^{3/2}}{3 d e}+\frac{2 a \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.101375, size = 56, normalized size = 0.89 \[ -\frac{2 \sqrt{e \cos (c+d x)} \left (b \cos ^{\frac{3}{2}}(c+d x)-3 a E\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{3 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.753, size = 123, normalized size = 2. \begin{align*}{\frac{2\,e}{3\,d} \left ( -4\,b \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+4\,b \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-b\sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) \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 \sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\right )}\,{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 (\sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\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 \sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]