3.10 \(\int (a+b \tan (c+d x^2))^2 \, dx\)

Optimal. Leaf size=16 \[ \text{Unintegrable}\left (\left (a+b \tan \left (c+d x^2\right )\right )^2,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0048063, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [A]  time = 1.9522, size = 0, normalized size = 0. \[ \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Tan[c + d*x^2])^2,x]

[Out]

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

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Maple [A]  time = 0.156, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\tan \left ( d{x}^{2}+c \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} x - \frac{b^{2} d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + b^{2} d x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, b^{2} d x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) + b^{2} d x^{2} - b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) - \frac{{\left (d x \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d x \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d x \cos \left (2 \, d x^{2} + 2 \, c\right ) + d x\right )}{\left (4 \, a d \int \frac{x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )}{x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) + x^{2}}\,{d x} + b \int \frac{\sin \left (2 \, d x^{2} + 2 \, c\right )}{x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + x^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, x^{2} \cos \left (2 \, d x^{2} + 2 \, c\right ) + x^{2}}\,{d x}\right )} b}{d}}{d x \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d x \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d x \cos \left (2 \, d x^{2} + 2 \, c\right ) + d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} \tan \left (d x^{2} + c\right )^{2} + 2 \, a b \tan \left (d x^{2} + c\right ) + a^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (c + d x^{2} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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