3.3.85 \(\int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\) [285]

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))},x\right ) \]

[Out]

Unintegrable(sec(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x)

________________________________________________________________________________________

Rubi [A]
time = 0.05, 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 {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sec[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]

[Out]

Defer[Int][Sec[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx &=\int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 23.85, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sec[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]

[Out]

Integrate[Sec[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]

________________________________________________________________________________________

Maple [A]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\sec ^{3}\left (d x +c \right )}{\left (f x +e \right ) \left (a +a \sin \left (d x +c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x)

[Out]

int(sec(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x)

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

________________________________________________________________________________________

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(sec(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

integral(sec(d*x + c)^3/(a*f*x + a*e + (a*f*x + a*e)*sin(d*x + c)), x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{e \sin {\left (c + d x \right )} + e + f x \sin {\left (c + d x \right )} + f x}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3/(f*x+e)/(a+a*sin(d*x+c)),x)

[Out]

Integral(sec(c + d*x)**3/(e*sin(c + d*x) + e + f*x*sin(c + d*x) + f*x), x)/a

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^3*(e + f*x)*(a + a*sin(c + d*x))),x)

[Out]

int(1/(cos(c + d*x)^3*(e + f*x)*(a + a*sin(c + d*x))), x)

________________________________________________________________________________________