\(\int \frac {(a+b \sec (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\) [365]

Optimal result

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

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

[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 \frac {(a+b \sec (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 10.64 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b \sec (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {(a+b \sec (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \]

[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 = 1.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[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 = 0.44 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\tan \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[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 [N/A]

Not integrable

Time = 52.60 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \sec (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \sec {\left (c + d x \right )}\right )^{n}}{\tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.97 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\tan \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[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 = 0.56 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\tan \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]

[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 = 20.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b \sec (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{{\mathrm {tan}\left (c+d\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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