Optimal. Leaf size=36 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d \sqrt {a+b}} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3181, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d \sqrt {a+b}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 3181
Rubi steps
\begin {align*} \int \frac {1}{a+b \sin ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {a+b} d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 36, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d \sqrt {a+b}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 236, normalized size = 6.56 \[ \left [-\frac {\sqrt {-a^{2} - a b} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{3} - {\left (a + b\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} - a b} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right )}{4 \, {\left (a^{2} + a b\right )} d}, -\frac {\arctan \left (\frac {{\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b}{2 \, \sqrt {a^{2} + a b} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right )}{2 \, \sqrt {a^{2} + a b} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 64, normalized size = 1.78 \[ \frac {\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )}{\sqrt {a^{2} + a b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 30, normalized size = 0.83 \[ \frac {\arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{d \sqrt {a \left (a +b \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 29, normalized size = 0.81 \[ \frac {\arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.53, size = 33, normalized size = 0.92 \[ \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a+b\right )}{\sqrt {a^2+b\,a}}\right )}{d\,\sqrt {a^2+b\,a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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