Optimal. Leaf size=61 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^2}+\frac{(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac{\tan (x) \sec (x)}{2 (a+b)} \]
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Rubi [A] time = 0.0867202, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3190, 414, 522, 206, 205} \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^2}+\frac{(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac{\tan (x) \sec (x)}{2 (a+b)} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 414
Rule 522
Rule 206
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^3(x)}{a+b \sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\sin (x)\right )\\ &=\frac{\sec (x) \tan (x)}{2 (a+b)}+\frac{\operatorname{Subst}\left (\int \frac{a+2 b+b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\sin (x)\right )}{2 (a+b)}\\ &=\frac{\sec (x) \tan (x)}{2 (a+b)}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sin (x)\right )}{(a+b)^2}+\frac{(a+3 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (x)\right )}{2 (a+b)^2}\\ &=\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^2}+\frac{(a+3 b) \tanh ^{-1}(\sin (x))}{2 (a+b)^2}+\frac{\sec (x) \tan (x)}{2 (a+b)}\\ \end{align*}
Mathematica [B] time = 0.315132, size = 147, normalized size = 2.41 \[ \frac{\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \csc (x)}{\sqrt{b}}\right )}{\sqrt{a}}+\frac{a+b}{\left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )^2}-\frac{a+b}{\left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^2}-2 (a+3 b) \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+2 (a+3 b) \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )}{4 (a+b)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 112, normalized size = 1.8 \begin{align*} -{\frac{1}{ \left ( 4\,a+4\,b \right ) \left ( -1+\sin \left ( x \right ) \right ) }}-{\frac{\ln \left ( -1+\sin \left ( x \right ) \right ) a}{4\, \left ( a+b \right ) ^{2}}}-{\frac{3\,\ln \left ( -1+\sin \left ( x \right ) \right ) b}{4\, \left ( a+b \right ) ^{2}}}-{\frac{1}{ \left ( 4\,a+4\,b \right ) \left ( 1+\sin \left ( x \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( x \right ) \right ) a}{4\, \left ( a+b \right ) ^{2}}}+{\frac{3\,\ln \left ( 1+\sin \left ( x \right ) \right ) b}{4\, \left ( a+b \right ) ^{2}}}+{\frac{{b}^{2}}{ \left ( a+b \right ) ^{2}}\arctan \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 2.80953, size = 556, normalized size = 9.11 \begin{align*} \left [\frac{2 \, b \sqrt{-\frac{b}{a}} \cos \left (x\right )^{2} \log \left (-\frac{b \cos \left (x\right )^{2} - 2 \, a \sqrt{-\frac{b}{a}} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right ) +{\left (a + 3 \, b\right )} \cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) -{\left (a + 3 \, b\right )} \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \,{\left (a + b\right )} \sin \left (x\right )}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (x\right )^{2}}, \frac{4 \, b \sqrt{\frac{b}{a}} \arctan \left (\sqrt{\frac{b}{a}} \sin \left (x\right )\right ) \cos \left (x\right )^{2} +{\left (a + 3 \, b\right )} \cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) -{\left (a + 3 \, b\right )} \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \,{\left (a + b\right )} \sin \left (x\right )}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (x\right )^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{3}{\left (x \right )}}{a + b \sin ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14683, size = 138, normalized size = 2.26 \begin{align*} \frac{b^{2} \arctan \left (\frac{b \sin \left (x\right )}{\sqrt{a b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{a b}} + \frac{{\left (a + 3 \, b\right )} \log \left (\sin \left (x\right ) + 1\right )}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac{{\left (a + 3 \, b\right )} \log \left (-\sin \left (x\right ) + 1\right )}{4 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac{\sin \left (x\right )}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )}{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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