\(\int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx\) [301]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=-\text {Int}\left (\frac {\sec (a+b x)}{c+d x},x\right )+\text {Int}\left (\frac {\sec ^3(a+b x)}{c+d x},x\right ) \]

[Out]

-Unintegrable(sec(b*x+a)/(d*x+c),x)+Unintegrable(sec(b*x+a)^3/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 32.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int \frac {\tan ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.05 (sec) , antiderivative size = 1331, normalized size of antiderivative = 60.50 \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

((d*cos(3*b*x + 3*a) + d*cos(b*x + a) + (b*d*x + b*c)*sin(3*b*x + 3*a) - (b*d*x + b*c)*sin(b*x + a))*cos(4*b*x
+ 4*a) + (2*d*cos(2*b*x + 2*a) - 2*(b*d*x + b*c)*sin(2*b*x + 2*a) + d)*cos(3*b*x + 3*a) + 2*(d*cos(b*x + a) -
(b*d*x + b*c)*sin(b*x + a))*cos(2*b*x + 2*a) + d*cos(b*x + a) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*d
^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a)^
2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b
*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^2*x^2 + 2*b
^2*c*d*x + b^2*c^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) + 4*(b^2*d^2*x
^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*integrate(((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(2*b
*x + 2*a)*cos(b*x + a) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*sin(2*b*x + 2*a)*sin(b*x + a) + (b^2*d^
2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(b*x + a))/(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3
+ (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^
2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*
cos(2*b*x + 2*a)), x) - ((b*d*x + b*c)*cos(3*b*x + 3*a) - (b*d*x + b*c)*cos(b*x + a) - d*sin(3*b*x + 3*a) - d*
sin(b*x + a))*sin(4*b*x + 4*a) + (b*d*x + b*c + 2*(b*d*x + b*c)*cos(2*b*x + 2*a) + 2*d*sin(2*b*x + 2*a))*sin(3
*b*x + 3*a) + 2*((b*d*x + b*c)*cos(b*x + a) + d*sin(b*x + a))*sin(2*b*x + 2*a) - (b*d*x + b*c)*sin(b*x + a))/(
b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^
2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 + 4*(
b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^
2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2
*b*x + 2*a))*cos(4*b*x + 4*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))

Giac [N/A]

Not integrable

Time = 5.86 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int \frac {{\mathrm {tan}\left (a+b\,x\right )}^2}{\cos \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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