Optimal. Leaf size=24 \[ \text {Int}\left (\left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x),x\right ) \]
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Rubi [A]
time = 0.02, 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 \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx &=\int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \sin ^n(c+d x)\right )^p \tan (c+d x) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.36, size = 0, normalized size = 0.00 \[\int \left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )^{p} \tan \left (d x +c \right )\, dx\]
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 = 0.41, size = 23, normalized size = 0.96 \begin {gather*} {\rm integral}\left ({\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right ), 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 \left (a + b \sin ^{n}{\left (c + d x \right )}\right )^{p} \tan {\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 \mathrm {tan}\left (c+d\,x\right )\,{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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