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

Optimal. Leaf size=39 \[ \text {Int}\left (\frac {\sec ^3(a+b x)}{c+d x},x\right )-\text {Int}\left (\frac {\sec (a+b x)}{c+d x},x\right ) \]

[Out]

-Unintegrable(sec(b*x+a)/(d*x+c),x)+Unintegrable(sec(b*x+a)^3/(d*x+c),x)

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-Defer[Int][Sec[a + b*x]/(c + d*x), x] + Defer[Int][Sec[a + b*x]^3/(c + d*x), x]

Rubi steps

\begin {align*} \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx &=-\int \frac {\sec (a+b x)}{c+d x} \, dx+\int \frac {\sec ^3(a+b x)}{c+d x} \, dx\\ \end {align*}

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Mathematica [A]  time = 26.39, size = 0, normalized size = 0.00 \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )^{2}}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.60, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x +a \right ) \left (\tan ^{2}\left (b x +a \right )\right )}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sec(b*x+a)*tan(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)*tan(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 = -0.03 \[ \int \frac {{\mathrm {tan}\left (a+b\,x\right )}^2}{\cos \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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