Optimal. Leaf size=102 \[ \frac{2 a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 b}{3 d e (e \sin (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0707494, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2669, 2636, 2642, 2641} \[ \frac{2 a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 b}{3 d e (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2669
Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{a+b \cos (c+d x)}{(e \sin (c+d x))^{5/2}} \, dx &=-\frac{2 b}{3 d e (e \sin (c+d x))^{3/2}}+a \int \frac{1}{(e \sin (c+d x))^{5/2}} \, dx\\ &=-\frac{2 b}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac{a \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{3 e^2}\\ &=-\frac{2 b}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac{\left (a \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{3 e^2 \sqrt{e \sin (c+d x)}}\\ &=-\frac{2 b}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 d e^2 \sqrt{e \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.151196, size = 59, normalized size = 0.58 \[ -\frac{2 \left (a \cos (c+d x)+a \sin ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )+b\right )}{3 d e (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.694, size = 124, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( -{\frac{2\,b}{3\,e} \left ( e\sin \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{a}{3\,{e}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) } \left ( \sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) } \left ( \sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+2\,\sin \left ( dx+c \right ) \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \right ) } \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{b \cos \left (d x + c\right ) + a}{\left (e \sin \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 (b \cos \left (d x + c\right ) + a\right )} \sqrt{e \sin \left (d x + c\right )}}{{\left (e^{3} \cos \left (d x + c\right )^{2} - e^{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{b \cos \left (d x + c\right ) + a}{\left (e \sin \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]