\(\int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx\) [295]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.13 (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 = {} \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 8.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \tan \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\tan {\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 1090, normalized size of antiderivative = 49.55 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \tan \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

(4*(b*d*x + b*c)*cos(2*b*x + 2*a)^2 + 4*(b*d*x + b*c)*sin(2*b*x + 2*a)^2 + (2*(b*d*x + b*c)*cos(2*b*x + 2*a) -
d*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) + 2*(b*d*x + b*c)*cos(2*b*x + 2*a) + 2*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^
2*c^2*d^2 + (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^4*x^2 + 2*b^2*c*d^3*x +
b^2*c^2*d^2)*cos(2*b*x + 2*a)^2 + (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*sin(4*b*x + 4*a)^2 + 4*(b^2*d^4*
x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^
2*d^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 + 2*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^
2*c^2*d^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) + 4*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(2*b*x + 2*a)
)*integrate(sin(2*b*x + 2*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) +
(d*cos(2*b*x + 2*a) + 2*(b*d*x + b*c)*sin(2*b*x + 2*a) + d)*sin(4*b*x + 4*a) + d*sin(2*b*x + 2*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 = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \tan \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 24.83 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\mathrm {tan}\left (a+b\,x\right )}{{\cos \left (a+b\,x\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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