Optimal. Leaf size=165 \[ -\frac{2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 \left (3 a^2-2 b^2\right ) E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d e^4 \sqrt{\sin (c+d x)}}-\frac{2 a b}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.171197, antiderivative size = 165, 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, 2669, 2636, 2640, 2639} \[ -\frac{2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 \left (3 a^2-2 b^2\right ) E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d e^4 \sqrt{\sin (c+d x)}}-\frac{2 a b}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2691
Rule 2669
Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{7/2}} \, dx &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac{2 \int \frac{-\frac{3 a^2}{2}+b^2-\frac{1}{2} a b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac{2 a b}{5 d e^3 \sqrt{e \sin (c+d x)}}+\frac{\left (3 a^2-2 b^2\right ) \int \frac{1}{(e \sin (c+d x))^{3/2}} \, dx}{5 e^2}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac{2 a b}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{\left (3 a^2-2 b^2\right ) \int \sqrt{e \sin (c+d x)} \, dx}{5 e^4}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac{2 a b}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{\left (\left (3 a^2-2 b^2\right ) \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{5 e^4 \sqrt{\sin (c+d x)}}\\ &=-\frac{2 (b+a \cos (c+d x)) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}-\frac{2 a b}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 \left (3 a^2-2 b^2\right ) \cos (c+d x)}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 \left (3 a^2-2 b^2\right ) E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 d e^4 \sqrt{\sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.489102, size = 109, normalized size = 0.66 \[ -\frac{\left (7 a^2+2 b^2\right ) \cos (c+d x)-4 \left (3 a^2-2 b^2\right ) \sin ^{\frac{5}{2}}(c+d x) E\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )-3 a^2 \cos (3 (c+d x))+8 a b+2 b^2 \cos (3 (c+d x))}{10 d e (e \sin (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.971, size = 327, normalized size = 2. \begin{align*}{\frac{1}{d} \left ( -{\frac{4\,ab}{5\,e} \left ( e\sin \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}+{\frac{1}{5\,{e}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) } \left ( \left ( 6\,{a}^{2}-4\,{b}^{2} \right ) \sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+ \left ( -8\,{a}^{2}+2\,{b}^{2} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +6\, \left ( \sin \left ( dx+c \right ) \right ) ^{7/2}{\it EllipticE} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) \sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }{a}^{2}-4\, \left ( \sin \left ( dx+c \right ) \right ) ^{7/2}{\it EllipticE} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) \sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }{b}^{2}-3\, \left ( \sin \left ( dx+c \right ) \right ) ^{7/2}{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) \sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }{a}^{2}+2\, \left ( \sin \left ( dx+c \right ) \right ) ^{7/2}{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) \sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }{b}^{2} \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{{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac{7}{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^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}\right )} \sqrt{e \sin \left (d x + c\right )}}{e^{4} \cos \left (d x + c\right )^{4} - 2 \, e^{4} \cos \left (d x + c\right )^{2} + e^{4}}, 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 \cos \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]