3.6.81 \(\int (a+b \sin ^n(c+d x))^p \tan ^m(c+d x) \, dx\) [581]

Optimal. Leaf size=26 \[ \text {Int}\left (\left (a+b \sin ^n(c+d x)\right )^p \tan ^m(c+d x),x\right ) \]

[Out]

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

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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 \left (a+b \sin ^n(c+d x)\right )^p \tan ^m(c+d x) \, dx \end {gather*}

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*}

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Mathematica [A]
time = 6.88, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \sin ^n(c+d x)\right )^p \tan ^m(c+d x) \, dx \end {gather*}

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 = 1.30, size = 0, normalized size = 0.00 \[\int \left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )^{p} \left (\tan ^{m}\left (d x +c \right )\right )\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 = 0.41, size = 25, 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 )^{m}, x\right ) \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 )}^m\,{\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.

[In]

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

[Out]

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

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