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

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

[Out]

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

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Rubi [A]  time = 0.0045918, 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 = {} \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx \]

Verification is Not applicable to the result.

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

Mathematica [A]  time = 0.705486, size = 0, normalized size = 0. \[ \int \left (a+b \tan \left (c+d x^2\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[a + b*Tan[c + d*x^2],x]

[Out]

Integrate[a + b*Tan[c + d*x^2], x]

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Maple [A]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int a+b\tan \left ( d{x}^{2}+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} a x + 2 \, b \int \frac{\sin \left (2 \, d x^{2} + 2 \, c\right )}{\cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x^{2} + 2 \, c\right ) + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b*tan(d*x^2 + c) + a, 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 )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*tan(c + d*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b*tan(d*x^2 + c) + a, x)