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 = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, 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 12
Rule 205
Rule 3181
Rule 3184
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*}
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Mathematica [A] time = 0.41, 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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fricas [B] time = 0.47, size = 463, normalized size = 5.32 \[ \left [-\frac {4 \, {\left (a^{2} b + a b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left ({\left (2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - 3 \, a b - b^{2}\right )} \sqrt {-a^{2} - a b} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{3} - {\left (a + b\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} - a b} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right )}{8 \, {\left ({\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )}}, -\frac {2 \, {\left (a^{2} b + a b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left ({\left (2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - 3 \, a b - b^{2}\right )} \sqrt {a^{2} + a b} \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b}{2 \, \sqrt {a^{2} + a b} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right )}{4 \, {\left ({\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 113, normalized size = 1.30 \[ \frac {\frac {{\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )} {\left (2 \, a + b\right )}}{{\left (a^{2} + a b\right )}^{\frac {3}{2}}} + \frac {b \tan \left (d x + c\right )}{{\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )} {\left (a^{2} + a b\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 119, normalized size = 1.37 \[ \frac {b \tan \left (d x +c \right )}{2 d a \left (a +b \right ) \left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}+\frac {\arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{d \left (a +b \right ) \sqrt {a \left (a +b \right )}}+\frac {\arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right ) b}{2 d a \left (a +b \right ) \sqrt {a \left (a +b \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 89, normalized size = 1.02 \[ \frac {\frac {b \tan \left (d x + c\right )}{a^{3} + a^{2} b + {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \tan \left (d x + c\right )^{2}} + \frac {{\left (2 \, a + b\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} {\left (a^{2} + a b\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.43, size = 79, normalized size = 0.91 \[ \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+2\,b\right )}{2\,\sqrt {a}\,\sqrt {a+b}}\right )\,\left (2\,a+b\right )}{2\,a^{3/2}\,d\,{\left (a+b\right )}^{3/2}}+\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{2\,a\,d\,\left (\left (a+b\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )\,\left (a+b\right )} \]
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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