\(\int \frac {a+b \tan (c+d x^2)}{x} \, dx\) [5]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx=a \log (x)+b \text {Int}\left (\frac {\tan \left (c+d x^2\right )}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx=\int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx \]

[In]

Int[(a + b*Tan[c + d*x^2])/x,x]

[Out]

a*Log[x] + b*Defer[Int][Tan[c + d*x^2]/x, x]

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

Mathematica [N/A]

Not integrable

Time = 1.70 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx=\int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx \]

[In]

Integrate[(a + b*Tan[c + d*x^2])/x,x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx=\int { \frac {b \tan \left (d x^{2} + c\right ) + a}{x} \,d x } \]

[In]

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

[Out]

integral((b*tan(d*x^2 + c) + a)/x, x)

Sympy [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx=\int \frac {a + b \tan {\left (c + d x^{2} \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 4.38 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx=\int { \frac {b \tan \left (d x^{2} + c\right ) + a}{x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx=\int { \frac {b \tan \left (d x^{2} + c\right ) + a}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.76 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \tan \left (c+d x^2\right )}{x} \, dx=\int \frac {a+b\,\mathrm {tan}\left (d\,x^2+c\right )}{x} \,d x \]

[In]

int((a + b*tan(c + d*x^2))/x,x)

[Out]

int((a + b*tan(c + d*x^2))/x, x)