\(\int \frac {\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx\) [263]

Optimal result

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

[Out]

CannotIntegrate(sec(b*x+a)*tan(b*x+a)/(d*x+c),x)-cos(a-b*c/d)*Si(b*c/d+b*x)/d-Ci(b*c/d+b*x)*sin(a-b*c/d)/d

Rubi [N/A]

Not integrable

Time = 0.21 (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 = {} \[ \int \frac {\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int \frac {\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

-((CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d) - (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d + Defer[I
nt][(Sec[a + b*x]*Tan[a + b*x])/(c + d*x), x]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\sin (a+b x)}{c+d x} \, dx+\int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx \\ & = -\left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx\right )-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx+\int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx \\ & = -\frac {\operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d}+\int \frac {\sec (a+b x) \tan (a+b x)}{c+d x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int \frac {\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.63 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int \frac {\sin {\left (a + b x \right )} \tan ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.74 (sec) , antiderivative size = 1268, normalized size of antiderivative = 57.64 \[ \int \frac {\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int { \frac {\sin \left (b x + a\right ) \tan \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

1/2*(b*c*(I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) - I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*
d)/d) + b*c*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*sin(-(b*c - a*d
)/d) + (b*c*(I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) - I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c -
a*d)/d) + b*c*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*sin(-(b*c -
a*d)/d) + (b*d*(I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) - I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*
c - a*d)/d) + b*d*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*sin(-(b*c
- a*d)/d))*x)*cos(2*b*x + 2*a)^2 + (b*c*(I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) - I*exp_integral_e(1, -(I*b
*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d) + b*c*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*
d*x + I*b*c)/d))*sin(-(b*c - a*d)/d) + (b*d*(I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) - I*exp_integral_e(1, -(
I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d) + b*d*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I
*b*d*x + I*b*c)/d))*sin(-(b*c - a*d)/d))*x)*sin(2*b*x + 2*a)^2 + 4*d*sin(2*b*x + 2*a)*sin(b*x + a) + (b*d*(I*e
xp_integral_e(1, (I*b*d*x + I*b*c)/d) - I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d) + b*d*(
exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*sin(-(b*c - a*d)/d))*x - 2*(
b*c*(-I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) + I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d
) - b*c*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*sin(-(b*c - a*d)/d)
+ (b*d*(-I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) + I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*
d)/d) - b*d*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*sin(-(b*c - a*d
)/d))*x - 2*d*cos(b*x + a))*cos(2*b*x + 2*a) + 4*d*cos(b*x + a) + 4*(b*d^3*x + b*c*d^2 + (b*d^3*x + b*c*d^2)*c
os(2*b*x + 2*a)^2 + (b*d^3*x + b*c*d^2)*sin(2*b*x + 2*a)^2 + 2*(b*d^3*x + b*c*d^2)*cos(2*b*x + 2*a))*integrate
((cos(2*b*x + 2*a)*cos(b*x + a) + sin(2*b*x + 2*a)*sin(b*x + a) + cos(b*x + a))/(b*d^2*x^2 + 2*b*c*d*x + b*c^2
+ (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cos(2*b*x + 2*a)^2 + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sin(2*b*x + 2*a)^2 + 2
*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cos(2*b*x + 2*a)), x))/(b*d^2*x + b*c*d + (b*d^2*x + b*c*d)*cos(2*b*x + 2*a)^
2 + (b*d^2*x + b*c*d)*sin(2*b*x + 2*a)^2 + 2*(b*d^2*x + b*c*d)*cos(2*b*x + 2*a))

Giac [N/A]

Not integrable

Time = 1.57 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int { \frac {\sin \left (b x + a\right ) \tan \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 27.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sin (a+b x) \tan ^2(a+b x)}{c+d x} \, dx=\int \frac {\sin \left (a+b\,x\right )\,{\mathrm {tan}\left (a+b\,x\right )}^2}{c+d\,x} \,d x \]

[In]

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

[Out]

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