\(\int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx\) [219]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx=-\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d}+\text {Int}\left (\frac {\sec (a+b x)}{c+d x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.88 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

integral(-(cos(b*x + a)^2 - 1)*sec(b*x + a)/(d*x + c), x)

Sympy [N/A]

Not integrable

Time = 1.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {\sin ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 210, normalized size of antiderivative = 10.50 \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx=\int { \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 24.45 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^2}{\cos \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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