Optimal. Leaf size=53 \[ -\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{3/2} d \sqrt {a+b}}-\frac {\cot (c+d x)}{a d} \]
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Rubi [A] time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3187, 453, 205} \[ -\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{3/2} d \sqrt {a+b}}-\frac {\cot (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 453
Rule 3187
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{x^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\cot (c+d x)}{a d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{a d}\\ &=-\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{3/2} \sqrt {a+b} d}-\frac {\cot (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 53, normalized size = 1.00 \[ \frac {-\frac {b \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a+b}}-\sqrt {a} \cot (c+d x)}{a^{3/2} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 313, normalized size = 5.91 \[ \left [-\frac {\sqrt {-a^{2} - a b} b \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{3} - {\left (a + b\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} - a b} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) \sin \left (d x + c\right ) + 4 \, {\left (a^{2} + a b\right )} \cos \left (d x + c\right )}{4 \, {\left (a^{3} + a^{2} b\right )} d \sin \left (d x + c\right )}, \frac {\sqrt {a^{2} + a b} b \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b}{2 \, \sqrt {a^{2} + a b} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 2 \, {\left (a^{2} + a b\right )} \cos \left (d x + c\right )}{2 \, {\left (a^{3} + a^{2} b\right )} d \sin \left (d x + c\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 83, normalized size = 1.57 \[ -\frac {\frac {{\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )} b}{\sqrt {a^{2} + a b} a} + \frac {1}{a \tan \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 52, normalized size = 0.98 \[ -\frac {b \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{d a \sqrt {a \left (a +b \right )}}-\frac {1}{d a \tan \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 48, normalized size = 0.91 \[ -\frac {\frac {b \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a} + \frac {1}{a \tan \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.48, size = 45, normalized size = 0.85 \[ -\frac {\mathrm {cot}\left (c+d\,x\right )}{a\,d}-\frac {b\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\sqrt {a+b}}{\sqrt {a}}\right )}{a^{3/2}\,d\,\sqrt {a+b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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