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

Optimal. Leaf size=17 \[ \text {Int}\left (\left (a+b \tan \left (c+d x^2\right )\right )^2,x\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, 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 = {} \begin {gather*} \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx \end {gather*}

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*}

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Mathematica [A]
time = 1.92, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \tan \left (c+d x^2\right )\right )^2 \, dx \end {gather*}

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 = 0.20, size = 0, normalized size = 0.00 \[\int \left (a +b \tan \left (d \,x^{2}+c \right )\right )^{2}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (c + d x^{2} \right )}\right )^{2}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int {\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(c + d*x^2))^2,x)

[Out]

int((a + b*tan(c + d*x^2))^2, x)

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