Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)},x\right ) \]
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Rubi [A]
time = 0.03, 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 = {}
\begin {gather*} \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx &=\int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx\\ \end {align*}
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Mathematica [A]
time = 5.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^m(c+d x)}{a+b \sin ^n(c+d x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.43, size = 0, normalized size = 0.00 \[\int \frac {\tan ^{m}\left (d x +c \right )}{a +b \left (\sin ^{n}\left (d x +c \right )\right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 25, normalized size = 0.96 \begin {gather*} {\rm integral}\left (\frac {\tan \left (d x + c\right )^{m}}{b \sin \left (d x + c\right )^{n} + a}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{m}{\left (c + d x \right )}}{a + b \sin ^{n}{\left (c + d x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{a+b\,{\sin \left (c+d\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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