3.564 \(\int (a+b \sin ^4(c+d x))^p \tan ^m(c+d x) \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\tan ^m(c+d x) \left (a+b \sin ^4(c+d x)\right )^p,x\right ) \]

[Out]

Unintegrable((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^m,x)

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Rubi [A]  time = 0.04, 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 = {} \[ \int \left (a+b \sin ^4(c+d x)\right )^p \tan ^m(c+d x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^m,x]

[Out]

Defer[Int][(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^m, x]

Rubi steps

\begin {align*} \int \left (a+b \sin ^4(c+d x)\right )^p \tan ^m(c+d x) \, dx &=\int \left (a+b \sin ^4(c+d x)\right )^p \tan ^m(c+d x) \, dx\\ \end {align*}

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Mathematica [A]  time = 5.75, size = 0, normalized size = 0.00 \[ \int \left (a+b \sin ^4(c+d x)\right )^p \tan ^m(c+d x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^m,x]

[Out]

Integrate[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^m, x]

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fricas [A]  time = 2.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \tan \left (d x + c\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^m,x, algorithm="fricas")

[Out]

integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c)^m, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right )^{4} + a\right )}^{p} \tan \left (d x + c\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^m,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c)^4 + a)^p*tan(d*x + c)^m, x)

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maple [A]  time = 5.07, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (\sin ^{4}\left (d x +c \right )\right )\right )^{p} \left (\tan ^{m}\left (d x +c \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^m,x)

[Out]

int((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^m,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right )^{4} + a\right )}^{p} \tan \left (d x + c\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^m,x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c)^4 + a)^p*tan(d*x + c)^m, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^m\,{\left (b\,{\sin \left (c+d\,x\right )}^4+a\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^m*(a + b*sin(c + d*x)^4)^p,x)

[Out]

int(tan(c + d*x)^m*(a + b*sin(c + d*x)^4)^p, x)

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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.

[In]

integrate((a+b*sin(d*x+c)**4)**p*tan(d*x+c)**m,x)

[Out]

Timed out

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