Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2},x\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx &=\int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx\\ \end {align*}
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Mathematica [A] time = 24.11, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^m(c+d x)}{\left (a+b \sin ^n(c+d x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\tan \left (d x + c\right )^{m}}{b^{2} \sin \left (d x + c\right )^{2 \, n} + 2 \, a b \sin \left (d x + c\right )^{n} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{m}\left (d x +c \right )}{\left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (d x + c\right )^{m}}{{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^2} \,d x \]
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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