Optimal. Leaf size=37 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right )}{\sqrt{b} d \sqrt{a+b}} \]
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Rubi [A] time = 0.0395803, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3186, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right )}{\sqrt{b} d \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 208
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{a+b \sin ^2(c+d x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \cos (c+d x)}{\sqrt{a+b}}\right )}{\sqrt{b} \sqrt{a+b} d}\\ \end{align*}
Mathematica [C] time = 0.156883, size = 97, normalized size = 2.62 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b}-i \sqrt{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a-b}}\right )+\tan ^{-1}\left (\frac{\sqrt{b}+i \sqrt{a} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{-a-b}}\right )}{\sqrt{b} d \sqrt{-a-b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 29, normalized size = 0.8 \begin{align*} -{\frac{1}{d}{\it Artanh} \left ({b\cos \left ( dx+c \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77946, size = 273, normalized size = 7.38 \begin{align*} \left [\frac{\log \left (-\frac{b \cos \left (d x + c\right )^{2} - 2 \, \sqrt{a b + b^{2}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right )}{2 \, \sqrt{a b + b^{2}} d}, \frac{\sqrt{-a b - b^{2}} \arctan \left (\frac{\sqrt{-a b - b^{2}} \cos \left (d x + c\right )}{a + b}\right )}{{\left (a b + b^{2}\right )} d}\right ] \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 [A] time = 1.1264, size = 50, normalized size = 1.35 \begin{align*} \frac{\arctan \left (\frac{b \cos \left (d x + c\right )}{\sqrt{-a b - b^{2}}}\right )}{\sqrt{-a b - b^{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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