Optimal. Leaf size=239 \[ -\frac{5 e^3 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a^3 d}-\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.366482, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2680, 2685, 2677, 2775, 203, 2833, 63, 215} \[ -\frac{5 e^3 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a^3 d}-\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2680
Rule 2685
Rule 2677
Rule 2775
Rule 203
Rule 2833
Rule 63
Rule 215
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a+a \sin (c+d x))^{3/2}}-\frac{\left (5 e^2\right ) \int \frac{(e \cos (c+d x))^{3/2}}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a+a \sin (c+d x))^{3/2}}-\frac{5 e^3 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a^3 d}-\frac{\left (5 e^4\right ) \int \frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{e \cos (c+d x)}} \, dx}{2 a^3}\\ &=-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a+a \sin (c+d x))^{3/2}}-\frac{5 e^3 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a^3 d}-\frac{\left (5 e^4 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \int \frac{\sqrt{1+\cos (c+d x)}}{\sqrt{e \cos (c+d x)}} \, dx}{2 a^2 (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (5 e^4 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \int \frac{\sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx}{2 a^2 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a+a \sin (c+d x))^{3/2}}-\frac{5 e^3 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a^3 d}+\frac{\left (5 e^4 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{2 a^2 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac{\left (5 e^4 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+e x^2} \, dx,x,-\frac{\sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}}\right )}{a^2 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a+a \sin (c+d x))^{3/2}}-\frac{5 e^3 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a^3 d}-\frac{5 e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}}\right ) \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{d \left (a^3+a^3 \cos (c+d x)+a^3 \sin (c+d x)\right )}+\frac{\left (5 e^3 \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{e}}} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{a^2 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a+a \sin (c+d x))^{3/2}}-\frac{5 e^3 \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a^3 d}+\frac{5 e^{7/2} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right ) \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{d \left (a^3+a^3 \cos (c+d x)+a^3 \sin (c+d x)\right )}-\frac{5 e^{7/2} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}}\right ) \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{d \left (a^3+a^3 \cos (c+d x)+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.11486, size = 80, normalized size = 0.33 \[ -\frac{2^{3/4} \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{9/2} \, _2F_1\left (\frac{5}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{9 a^3 d e (\sin (c+d x)+1)^{11/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.117, size = 443, normalized size = 1.9 \begin{align*}{\frac{1}{4\,d \left ( 2\,\sin \left ( dx+c \right ) + \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2 \right ) } \left ( 5\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \sqrt{2}\sin \left ( dx+c \right ) -5\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) \sqrt{2}\sin \left ( dx+c \right ) +5\,\cos \left ( dx+c \right ) \sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}-5\,\cos \left ( dx+c \right ) \sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) +5\,\sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}-5\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) +4\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +36\,\cos \left ( dx+c \right ) \right ) \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{7}{2}}} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{-{\frac{5}{2}}}} \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 (e \cos \left (d x + c\right )\right )^{\frac{7}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
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(-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]