Optimal. Leaf size=191 \[ -\frac{a^{3/2} \left (A d (c+3 d)-B \left (3 c^2+3 c d-2 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{d^{5/2} f (c+d)^{3/2}}-\frac{a^2 (-A d+3 B c+2 B d) \cos (e+f x)}{d^2 f (c+d) \sqrt{a \sin (e+f x)+a}}+\frac{a (B c-A d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{d f (c+d) (c+d \sin (e+f x))} \]
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Rubi [A] time = 0.552218, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {2975, 2981, 2773, 208} \[ -\frac{a^{3/2} \left (A d (c+3 d)-B \left (3 c^2+3 c d-2 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{d^{5/2} f (c+d)^{3/2}}-\frac{a^2 (-A d+3 B c+2 B d) \cos (e+f x)}{d^2 f (c+d) \sqrt{a \sin (e+f x)+a}}+\frac{a (B c-A d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{d f (c+d) (c+d \sin (e+f x))} \]
Antiderivative was successfully verified.
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Rule 2975
Rule 2981
Rule 2773
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{(c+d \sin (e+f x))^2} \, dx &=\frac{a (B c-A d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{d (c+d) f (c+d \sin (e+f x))}+\frac{\int \frac{\sqrt{a+a \sin (e+f x)} \left (-\frac{1}{2} a (B c-3 A d-2 B d)+\frac{1}{2} a (3 B c-A d+2 B d) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{d (c+d)}\\ &=-\frac{a^2 (3 B c-A d+2 B d) \cos (e+f x)}{d^2 (c+d) f \sqrt{a+a \sin (e+f x)}}+\frac{a (B c-A d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{d (c+d) f (c+d \sin (e+f x))}+\frac{\left (a \left (A d (c+3 d)-B \left (3 c^2+3 c d-2 d^2\right )\right )\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{2 d^2 (c+d)}\\ &=-\frac{a^2 (3 B c-A d+2 B d) \cos (e+f x)}{d^2 (c+d) f \sqrt{a+a \sin (e+f x)}}+\frac{a (B c-A d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{d (c+d) f (c+d \sin (e+f x))}-\frac{\left (a^2 \left (A d (c+3 d)-B \left (3 c^2+3 c d-2 d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a c+a d-d x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{d^2 (c+d) f}\\ &=-\frac{a^{3/2} \left (A d (c+3 d)-B \left (3 c^2+3 c d-2 d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a+a \sin (e+f x)}}\right )}{d^{5/2} (c+d)^{3/2} f}-\frac{a^2 (3 B c-A d+2 B d) \cos (e+f x)}{d^2 (c+d) f \sqrt{a+a \sin (e+f x)}}+\frac{a (B c-A d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{d (c+d) f (c+d \sin (e+f x))}\\ \end{align*}
Mathematica [A] time = 4.88958, size = 381, normalized size = 1.99 \[ \frac{(a (\sin (e+f x)+1))^{3/2} \left (\frac{\left (A d (c+3 d)+B \left (-3 c^2-3 c d+2 d^2\right )\right ) \left (2 \log \left (\sqrt{d} \sqrt{c+d} \left (\tan ^2\left (\frac{1}{4} (e+f x)\right )+2 \tan \left (\frac{1}{4} (e+f x)\right )-1\right )+(c+d) \sec ^2\left (\frac{1}{4} (e+f x)\right )\right )-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+e+f x\right )}{(c+d)^{3/2}}+\frac{\left (B \left (3 c^2+3 c d-2 d^2\right )-A d (c+3 d)\right ) \left (2 \log \left (-\sec ^2\left (\frac{1}{4} (e+f x)\right ) \left (-\sqrt{d} \sqrt{c+d} \sin \left (\frac{1}{2} (e+f x)\right )+\sqrt{d} \sqrt{c+d} \cos \left (\frac{1}{2} (e+f x)\right )+c+d\right )\right )-2 \log \left (\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )+e+f x\right )}{(c+d)^{3/2}}-\frac{4 \sqrt{d} (d-c) (A d-B c) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{(c+d) (c+d \sin (e+f x))}+8 B \sqrt{d} \sin \left (\frac{1}{2} (e+f x)\right )-8 B \sqrt{d} \cos \left (\frac{1}{2} (e+f x)\right )\right )}{4 d^{5/2} f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.839, size = 592, normalized size = 3.1 \begin{align*}{\frac{a \left ( 1+\sin \left ( fx+e \right ) \right ) }{ \left ( c+d \right ){d}^{2} \left ( c+d\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) f}\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( -\sin \left ( fx+e \right ) d \left ( A{\it Artanh} \left ({d\sqrt{a-a\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{acd+a{d}^{2}}}}} \right ) acd+3\,A{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( fx+e \right ) }d}{\sqrt{acd+a{d}^{2}}}} \right ) a{d}^{2}-3\,B{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( fx+e \right ) }d}{\sqrt{acd+a{d}^{2}}}} \right ) a{c}^{2}-3\,B{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( fx+e \right ) }d}{\sqrt{acd+a{d}^{2}}}} \right ) acd+2\,B{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( fx+e \right ) }d}{\sqrt{acd+a{d}^{2}}}} \right ) a{d}^{2}+2\,B\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{a \left ( c+d \right ) d}c+2\,B\sqrt{a \left ( c+d \right ) d}\sqrt{a-a\sin \left ( fx+e \right ) }d \right ) -A{\it Artanh} \left ({d\sqrt{a-a\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{acd+a{d}^{2}}}}} \right ) a{c}^{2}d-3\,A{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( fx+e \right ) }d}{\sqrt{acd+a{d}^{2}}}} \right ) ac{d}^{2}+3\,B{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( fx+e \right ) }d}{\sqrt{acd+a{d}^{2}}}} \right ) a{c}^{3}+3\,B{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( fx+e \right ) }d}{\sqrt{acd+a{d}^{2}}}} \right ) a{c}^{2}d-2\,B{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( fx+e \right ) }d}{\sqrt{acd+a{d}^{2}}}} \right ) ac{d}^{2}+A\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{a \left ( c+d \right ) d}cd-A\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{a \left ( c+d \right ) d}{d}^{2}-3\,B\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{a \left ( c+d \right ) d}{c}^{2}-B\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{a \left ( c+d \right ) d}cd \right ){\frac{1}{\sqrt{a \left ( c+d \right ) d}}}{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 11.553, size = 3245, normalized size = 16.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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