3.1.16 \(\int \frac {1}{a+b \tan (c+d x^2)} \, dx\) [16]

Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{a+b \tan \left (c+d x^2\right )},x\right ) \]

[Out]

Unintegrable(1/(a+b*tan(d*x^2+c)),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 \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*Tan[c + d*x^2])^(-1),x]

[Out]

Defer[Int][(a + b*Tan[c + d*x^2])^(-1), x]

Rubi steps

\begin {align*} \int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx &=\int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*Tan[c + d*x^2])^(-1),x]

[Out]

Integrate[(a + b*Tan[c + d*x^2])^(-1), x]

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Maple [A]
time = 0.29, size = 0, normalized size = 0.00 \[\int \frac {1}{a +b \tan \left (d \,x^{2}+c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(d*x^2+c)),x)

[Out]

int(1/(a+b*tan(d*x^2+c)),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(1/(a+b*tan(d*x^2+c)),x, algorithm="maxima")

[Out]

(a*x + 2*(a^2*b + b^3)*integrate((2*a*b*cos(2*d*x^2 + 2*c) - (a^2 - b^2)*sin(2*d*x^2 + 2*c))/(a^4 + 2*a^2*b^2
+ b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x^2 + 2*c)^2 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x^2 + 2*c)^2 + 2*(a^4 -
b^4)*cos(2*d*x^2 + 2*c) + 4*(a^3*b + a*b^3)*sin(2*d*x^2 + 2*c)), x))/(a^2 + b^2)

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

[Out]

integral(1/(b*tan(d*x^2 + c) + a), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a + b \tan {\left (c + d x^{2} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x**2+c)),x)

[Out]

Integral(1/(a + b*tan(c + d*x**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(1/(a+b*tan(d*x^2+c)),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tan(c + d*x^2)),x)

[Out]

int(1/(a + b*tan(c + d*x^2)), x)

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