Optimal. Leaf size=87 \[ \frac{(2 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d (a+b)^{3/2}}+\frac{b \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )} \]
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Rubi [A] time = 0.0625083, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3184, 12, 3181, 205} \[ \frac{(2 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d (a+b)^{3/2}}+\frac{b \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 3184
Rule 12
Rule 3181
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx &=\frac{b \cos (c+d x) \sin (c+d x)}{2 a (a+b) d \left (a+b \sin ^2(c+d x)\right )}-\frac{\int \frac{-2 a-b}{a+b \sin ^2(c+d x)} \, dx}{2 a (a+b)}\\ &=\frac{b \cos (c+d x) \sin (c+d x)}{2 a (a+b) d \left (a+b \sin ^2(c+d x)\right )}+\frac{(2 a+b) \int \frac{1}{a+b \sin ^2(c+d x)} \, dx}{2 a (a+b)}\\ &=\frac{b \cos (c+d x) \sin (c+d x)}{2 a (a+b) d \left (a+b \sin ^2(c+d x)\right )}+\frac{(2 a+b) \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{2 a (a+b) d}\\ &=\frac{(2 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} (a+b)^{3/2} d}+\frac{b \cos (c+d x) \sin (c+d x)}{2 a (a+b) d \left (a+b \sin ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.41682, size = 84, normalized size = 0.97 \[ \frac{\frac{(2 a+b) \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{(a+b)^{3/2}}+\frac{\sqrt{a} b \sin (2 (c+d x))}{(a+b) (2 a-b \cos (2 (c+d x))+b)}}{2 a^{3/2} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 119, normalized size = 1.4 \begin{align*}{\frac{b\tan \left ( dx+c \right ) }{2\,da \left ( a+b \right ) \left ( a \left ( \tan \left ( dx+c \right ) \right ) ^{2}+ \left ( \tan \left ( dx+c \right ) \right ) ^{2}b+a \right ) }}+{\frac{1}{d \left ( a+b \right ) }\arctan \left ({ \left ( a+b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}}+{\frac{b}{2\,da \left ( a+b \right ) }\arctan \left ({ \left ( a+b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \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 [B] time = 1.88266, size = 1060, normalized size = 12.18 \begin{align*} \left [-\frac{4 \,{\left (a^{2} b + a b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left ({\left (2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - 3 \, a b - b^{2}\right )} \sqrt{-a^{2} - a b} \log \left (\frac{{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{3} -{\left (a + b\right )} \cos \left (d x + c\right )\right )} \sqrt{-a^{2} - a b} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \,{\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right )}{8 \,{\left ({\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} d \cos \left (d x + c\right )^{2} -{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )}}, -\frac{2 \,{\left (a^{2} b + a b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left ({\left (2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - 3 \, a b - b^{2}\right )} \sqrt{a^{2} + a b} \arctan \left (\frac{{\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right )}{4 \,{\left ({\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} d \cos \left (d x + c\right )^{2} -{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )}}\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 = 1.10842, size = 153, normalized size = 1.76 \begin{align*} \frac{\frac{{\left (\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac{a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt{a^{2} + a b}}\right )\right )}{\left (2 \, a + b\right )}}{{\left (a^{2} + a b\right )}^{\frac{3}{2}}} + \frac{b \tan \left (d x + c\right )}{{\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}{\left (a^{2} + a b\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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