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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.70, 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 = {} \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 15.43, size = 0, normalized size = 0.00 \[ \int \csc ^3(c+d x) (a+b \tan (c+d x))^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*tan(d*x + c) + a)^n*csc(d*x + c)^3, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.75, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{3}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tan(c + d*x))**n*csc(c + d*x)**3, x)

________________________________________________________________________________________