3.581 \(\int (a+b \sin ^n(c+d x))^p \tan ^m(c+d x) \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\tan ^m(c+d x) \left (a+b \sin ^n(c+d x)\right )^p,x\right ) \]

[Out]

Unintegrable[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^m, x]

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Rubi [A]  time = 0.0521793, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^m(c+d x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^m,x]

[Out]

Defer[Int][(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^m, x]

Rubi steps

\begin{align*} \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^m(c+d x) \, dx &=\int \left (a+b \sin ^n(c+d x)\right )^p \tan ^m(c+d x) \, dx\\ \end{align*}

Mathematica [A]  time = 3.8874, size = 0, normalized size = 0. \[ \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^m(c+d x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^m,x]

[Out]

Integrate[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^m, x]

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Maple [A]  time = 2.234, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ( \sin \left ( dx+c \right ) \right ) ^{n} \right ) ^{p} \left ( \tan \left ( dx+c \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^m,x)

[Out]

int((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^m,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^m,x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^m, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^m,x, algorithm="fricas")

[Out]

integral((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)**n)**p*tan(d*x+c)**m,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \tan \left (d x + c\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^m,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^m, x)