\(\int (a+b \tan (c+d x^2))^2 \, dx\) [10]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx=\text {Int}\left (\left (a+b \tan \left (c+d x^2\right )\right )^2,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 14, 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 = {} \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx=\int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx \]

[In]

Int[(a + b*Tan[c + d*x^2])^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.89 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx=\int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx \]

[In]

Integrate[(a + b*Tan[c + d*x^2])^2,x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.29 \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx=\int \left (a + b \tan {\left (c + d x^{2} \right )}\right )^{2}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 288, normalized size of antiderivative = 20.57 \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.73 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx=\int {\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}^2 \,d x \]

[In]

int((a + b*tan(c + d*x^2))^2,x)

[Out]

int((a + b*tan(c + d*x^2))^2, x)