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

Optimal. Leaf size=69 \[ \text {Int}\left (\frac {\sec (a+b x)}{c+d x},x\right )-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d} \]

[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)

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Rubi [A]  time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx &=-\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 ) \text {Ci}\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*}

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Mathematica [A]  time = 6.04, size = 0, normalized size = 0.00 \[ \int \frac {\sin (a+b x) \tan (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sec \left (b x + a\right )}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x + a\right ) \sin \left (b x + a\right )^{2}}{d x + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x +a \right ) \left (\sin ^{2}\left (b x +a \right )\right )}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+b\,x\right )}^2}{\cos \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}}{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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