3.88 \(\int \sin ^3(c+d x) (a+b \tan (c+d x))^n \, dx\)

Optimal. Leaf size=24 \[ \text {Int}\left (\sin ^3(c+d x) (a+b \tan (c+d x))^n,x\right ) \]

[Out]

CannotIntegrate(sin(d*x+c)^3*(a+b*tan(d*x+c))^n,x)

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Rubi [A]  time = 1.98, 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 = {} \[ \int \sin ^3(c+d x) (a+b \tan (c+d x))^n \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]

[Out]

Defer[Int][Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n, x]

Rubi steps

\begin {align*} \int \sin ^3(c+d x) (a+b \tan (c+d x))^n \, dx &=\int \sin ^3(c+d x) (a+b \tan (c+d x))^n \, dx\\ \end {align*}

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Mathematica [A]  time = 3.21, size = 0, normalized size = 0.00 \[ \int \sin ^3(c+d x) (a+b \tan (c+d x))^n \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]

[Out]

Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n, x]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (d x + c\right )^{2} - 1\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3*(a+b*tan(d*x+c))^n,x, algorithm="fricas")

[Out]

integral(-(cos(d*x + c)^2 - 1)*(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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3*(a+b*tan(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*tan(d*x + c) + a)^n*sin(d*x + c)^3, x)

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maple [A]  time = 1.87, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{3}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^3*(a+b*tan(d*x+c))^n,x)

[Out]

int(sin(d*x+c)^3*(a+b*tan(d*x+c))^n,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3*(a+b*tan(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^n*sin(d*x + c)^3, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\sin \left (c+d\,x\right )}^3\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^3*(a + b*tan(c + d*x))^n,x)

[Out]

int(sin(c + d*x)^3*(a + b*tan(c + d*x))^n, 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(sin(d*x+c)**3*(a+b*tan(d*x+c))**n,x)

[Out]

Timed out

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