\(\int (a+b \tan (c+d x^2)) \, dx\) [4]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx=a x+b \text {Int}\left (\tan \left (c+d x^2\right ),x\right ) \]

[Out]

a*x+b*Unintegrable(tan(d*x^2+c),x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 12, 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 = {} \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx=\int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx \]

[In]

Int[a + b*Tan[c + d*x^2],x]

[Out]

a*x + b*Defer[Int][Tan[c + d*x^2], x]

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

Mathematica [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx=\int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx \]

[In]

Integrate[a + b*Tan[c + d*x^2],x]

[Out]

Integrate[a + b*Tan[c + d*x^2], x]

Maple [N/A] (verified)

Not integrable

Time = 0.14 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx=\int { b \tan \left (d x^{2} + c\right ) + a \,d x } \]

[In]

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

[Out]

integral(b*tan(d*x^2 + c) + a, x)

Sympy [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx=\int \left (a + b \tan {\left (c + d x^{2} \right )}\right )\, dx \]

[In]

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

[Out]

Integral(a + b*tan(c + d*x**2), x)

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 5.33 \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx=\int { b \tan \left (d x^{2} + c\right ) + a \,d x } \]

[In]

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

[Out]

a*x + 2*b*integrate(sin(2*d*x^2 + 2*c)/(cos(2*d*x^2 + 2*c)^2 + sin(2*d*x^2 + 2*c)^2 + 2*cos(2*d*x^2 + 2*c) + 1
), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx=\int { b \tan \left (d x^{2} + c\right ) + a \,d x } \]

[In]

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

[Out]

integrate(b*tan(d*x^2 + c) + a, x)

Mupad [N/A]

Not integrable

Time = 4.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx=\int a+b\,\mathrm {tan}\left (d\,x^2+c\right ) \,d x \]

[In]

int(a + b*tan(c + d*x^2),x)

[Out]

int(a + b*tan(c + d*x^2), x)