\(\int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx\) [362]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx=\text {Int}\left ((a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x),x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^n*tan(d*x+c)^(3/2),x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx=\int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx \]

[In]

Int[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^(3/2),x]

[Out]

Defer[Int][(a + b*Sec[c + d*x])^n*Tan[c + d*x]^(3/2), x]

Rubi steps \begin{align*} \text {integral}& = \int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.60 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx=\int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx \]

[In]

Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^(3/2),x]

[Out]

Integrate[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^(3/2), x]

Maple [N/A] (verified)

Not integrable

Time = 1.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \left (a +b \sec \left (d x +c \right )\right )^{n} \tan \left (d x +c \right )^{\frac {3}{2}}d x\]

[In]

int((a+b*sec(d*x+c))^n*tan(d*x+c)^(3/2),x)

[Out]

int((a+b*sec(d*x+c))^n*tan(d*x+c)^(3/2),x)

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^n*tan(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

integral((b*sec(d*x + c) + a)^n*tan(d*x + c)^(3/2), x)

Sympy [F(-1)]

Timed out. \[ \int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))**n*tan(d*x+c)**(3/2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.92 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^n*tan(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^n*tan(d*x + c)^(3/2), x)

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^n*tan(d*x+c)^(3/2),x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^n*tan(d*x + c)^(3/2), x)

Mupad [N/A]

Not integrable

Time = 19.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (a+b \sec (c+d x))^n \tan ^{\frac {3}{2}}(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

[In]

int(tan(c + d*x)^(3/2)*(a + b/cos(c + d*x))^n,x)

[Out]

int(tan(c + d*x)^(3/2)*(a + b/cos(c + d*x))^n, x)