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

Optimal. Leaf size=22 \[ \text {Int}\left (\sin (c+d x) (a+b \tan (c+d x))^n,x\right ) \]

[Out]

CannotIntegrate(sin(d*x+c)*(a+b*tan(d*x+c))^n,x)

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Rubi [A]  time = 0.86, 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 = {} \[ \int \sin (c+d x) (a+b \tan (c+d x))^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.25, size = 0, normalized size = 0.00 \[ \int \sin (c+d x) (a+b \tan (c+d x))^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.46, size = 0, normalized size = 0.00 \[ \int \sin \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \sin \left (c+d\,x\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tan(c + d*x))**n*sin(c + d*x), x)

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