\(\int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx\) [296]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.18 (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 = {} \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.79 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \tan \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\tan {\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.71 (sec) , antiderivative size = 1396, normalized size of antiderivative = 63.45 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \tan \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

2*(2*(b*d*x + b*c)*cos(2*b*x + 2*a)^2 + 2*(b*d*x + b*c)*sin(2*b*x + 2*a)^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) + (b*d*x + b*c)*cos(2*b*x + 2*a) + 3*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*
b^2*c^2*d^3*x + b^2*c^3*d^2 + (b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*cos(4*b*x + 4*a)
^2 + 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*cos(2*b*x + 2*a)^2 + (b^2*d^5*x^3 + 3*b
^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*sin(4*b*x + 4*a)^2 + 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^
2*d^3*x + b^2*c^3*d^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x
+ b^2*c^3*d^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2 + 2*(b^2*
d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) + 4*(b^2*d^5*x^3
+ 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*cos(2*b*x + 2*a))*integrate(sin(2*b*x + 2*a)/(b^2*d^4*x^4
+ 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d
^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*cos(2*b*x + 2*a)^2 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*
b^2*c^3*d*x + b^2*c^4)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d
*x + b^2*c^4)*cos(2*b*x + 2*a)), x) + (d*cos(2*b*x + 2*a) + (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^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(4*b*x + 4*a)^2 + 4*(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(4*b*x + 4*a)^2 + 4*(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(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(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 + 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))*cos(4*b*x + 4*a) + 4*
(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))

Giac [N/A]

Not integrable

Time = 6.74 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\sec \left (b x + a\right )^{2} \tan \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 26.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^2(a+b x) \tan (a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {tan}\left (a+b\,x\right )}{{\cos \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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