Optimal. Leaf size=105 \[ -\frac{a \cos (e+f x)}{f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{d} f (c+d)^{3/2}} \]
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Rubi [A] time = 0.185632, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2772, 2773, 208} \[ -\frac{a \cos (e+f x)}{f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{d} f (c+d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2772
Rule 2773
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^2} \, dx &=-\frac{a \cos (e+f x)}{(c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac{\int \frac{\sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{2 (c+d)}\\ &=-\frac{a \cos (e+f x)}{(c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac{a \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 )}{(c+d) f}\\ &=-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{c+d} \sqrt{a+a \sin (e+f x)}}\right )}{\sqrt{d} (c+d)^{3/2} f}-\frac{a \cos (e+f x)}{(c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))}\\ \end{align*}
Mathematica [C] time = 5.83129, size = 871, normalized size = 8.3 \[ \frac{\left (\frac{1}{4}+\frac{i}{4}\right ) \sqrt{a (\sin (e+f x)+1)} \left (\frac{\left (\cos \left (\frac{e}{2}\right )+i \sin \left (\frac{e}{2}\right )\right ) \left ((-1+i) x \cos (e)+(1+i) x \sin (e)+\frac{\text{RootSum}\left [d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\& ,\frac{-\sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3-2 i \sqrt{d} \sqrt{c+d} e^{i e} \log \left (e^{\frac{i f x}{2}}-\text{$\#$1}\right ) \text{$\#$1}^3+\frac{(1-i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2+2 i) c \log \left (e^{\frac{i f x}{2}}-\text{$\#$1}\right ) \text{$\#$1}^2}{\sqrt{e^{-i e}}}-i \sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 \sqrt{d} \sqrt{c+d} \log \left (e^{\frac{i f x}{2}}-\text{$\#$1}\right ) \text{$\#$1}+(1+i) d \sqrt{e^{-i e}} f x-(2-2 i) d \sqrt{e^{-i e}} \log \left (e^{\frac{i f x}{2}}-\text{$\#$1}\right )}{d-i c e^{i e} \text{$\#$1}^2}\& \right ] (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}{4 f}\right )}{\sqrt{d} (c+d)^{3/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}+\frac{\left (\cos \left (\frac{e}{2}\right )+i \sin \left (\frac{e}{2}\right )\right ) \left ((1-i) x \cos (e)-(1+i) x \sin (e)+\frac{\text{RootSum}\left [d e^{2 i e} \text{$\#$1}^4+2 i c e^{i e} \text{$\#$1}^2-d\& ,\frac{-i \sqrt{d} \sqrt{c+d} e^{i e} f x \text{$\#$1}^3+2 \sqrt{d} \sqrt{c+d} e^{i e} \log \left (e^{\frac{i f x}{2}}-\text{$\#$1}\right ) \text{$\#$1}^3-\frac{(1+i) c f x \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\frac{(2-2 i) c \log \left (e^{\frac{i f x}{2}}-\text{$\#$1}\right ) \text{$\#$1}^2}{\sqrt{e^{-i e}}}+\sqrt{d} \sqrt{c+d} f x \text{$\#$1}+2 i \sqrt{d} \sqrt{c+d} \log \left (e^{\frac{i f x}{2}}-\text{$\#$1}\right ) \text{$\#$1}+(1-i) d \sqrt{e^{-i e}} f x+(2+2 i) d \sqrt{e^{-i e}} \log \left (e^{\frac{i f x}{2}}-\text{$\#$1}\right )}{d-i c e^{i e} \text{$\#$1}^2}\& \right ] \sqrt{\cos (e)-i \sin (e)} (-i \cos (e)+\sin (e)-1)}{4 f}\right )}{\sqrt{d} (c+d)^{3/2} (\cos (e)+i (\sin (e)-1)) \sqrt{\cos (e)-i \sin (e)}}-\frac{(2-2 i) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{(c+d) f (c+d \sin (e+f x))}\right )}{\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.078, size = 155, normalized size = 1.5 \begin{align*} -{\frac{1+\sin \left ( fx+e \right ) }{ \left ( c+d \right ) \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 ({\it Artanh} \left ({d\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{\frac{1}{\sqrt{a \left ( c+d \right ) d}}}} \right ) \sin \left ( fx+e \right ) ad+{\it Artanh} \left ({d\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{\frac{1}{\sqrt{a \left ( c+d \right ) d}}}} \right ) ac+\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }\sqrt{a \left ( c+d \right ) d} \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{\sqrt{a \sin \left (f x + e\right ) + a}}{{\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 = 2.61356, size = 1874, normalized size = 17.85 \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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