3.3.14 \(\int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\) [214]
Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \]
[Out]
Unintegrable(csc(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x)
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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 {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[Csc[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]
[Out]
Defer[Int][Csc[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx &=\int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\\ \end {align*}
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Mathematica [A]
time = 110.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[Csc[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]
[Out]
Integrate[Csc[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]
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Maple [A]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\csc ^{3}\left (d x +c \right )}{\left (f x +e \right )^{2} \left (a +a \sin \left (d x +c \right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x)
[Out]
int(csc(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+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(csc(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
(2*f*cos(4*d*x + 4*c)^2 + 4*f*cos(3*d*x + 3*c)^2 + 4*f*cos(2*d*x + 2*c)^2 + 2*f*cos(d*x + c)^2 + 2*f*sin(4*d*x
+ 4*c)^2 + 4*f*sin(3*d*x + 3*c)^2 + 4*f*sin(2*d*x + 2*c)^2 + 2*f*sin(d*x + c)^2 + (4*d*f*x + 4*d*e + 3*(d*f*x
+ d*e)*cos(4*d*x + 4*c) - 2*f*cos(3*d*x + 3*c) - 5*(d*f*x + d*e)*cos(2*d*x + 2*c) + 2*f*cos(d*x + c) - 2*f*si
n(4*d*x + 4*c) - 3*(d*f*x + d*e)*sin(3*d*x + 3*c) + 2*f*sin(2*d*x + 2*c) + (d*f*x + d*e)*sin(d*x + c))*cos(5*d
*x + 5*c) - (3*(d*f*x + d*e)*cos(3*d*x + 3*c) + 6*f*cos(2*d*x + 2*c) - 2*(d*f*x + d*e)*cos(d*x + c) + 6*f*sin(
3*d*x + 3*c) - (d*f*x + d*e)*sin(2*d*x + 2*c) - 4*f*sin(d*x + c) - 2*f)*cos(4*d*x + 4*c) - (5*d*f*x + 5*d*e -
4*(d*f*x + d*e)*cos(2*d*x + 2*c) + 6*f*cos(d*x + c) + 8*f*sin(2*d*x + 2*c) - (d*f*x + d*e)*sin(d*x + c))*cos(3
*d*x + 3*c) - (3*(d*f*x + d*e)*cos(d*x + c) + 6*f*sin(d*x + c) + 2*f)*cos(2*d*x + 2*c) + 3*(d*f*x + d*e)*cos(d
*x + c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^
2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(5*d*x + 5*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a
*d^2*e^3)*cos(4*d*x + 4*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(3*d*x +
3*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c)^2 + (a*d^2*f^3*
x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3
*a*d^2*e^2*f*x + a*d^2*e^3)*sin(5*d*x + 5*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*
e^3)*sin(4*d*x + 4*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(3*d*x + 3*c)
^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)*sin(2*d*x + 2*c) + 4*(a*
d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e
*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c)^2 - 2*(2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2
*f*x + a*d^2*e^3)*cos(3*d*x + 3*c) - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x
+ c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(4*d*x + 4*c) - 2*(a*d^2*f^3*x^3
+ 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) + 2*(a*d^2*f^3*x^3 + 3*a
*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*
e^3)*cos(2*d*x + 2*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(3*d*x + 3*c) +
(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*cos(4*d*x + 4*c) - 4*((a*d^2*
f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c) + 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2
+ 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*cos(3*d*x + 3*c) - 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*
a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*co
s(2*d*x + 2*c) + 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2
*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(4*d*x + 4*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^
2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(
3*d*x + 3*c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*sin(5*d*x + 5*c
) + 2*(2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(3*d*x + 3*c) - (a*d^2*f^3*x^3 +
3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^
2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c) - 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2
*f*x + a*d^2*e^3 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c) + (a*d
^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*sin(3*d*x + 3*c) + 2*(a*d^2*f^3*x^
3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*integrate(1/2*(3*d^2*f^2*x^2 + 3*d^2*e^2 +
4*d*e*f + 6*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*sin(d*x + c)/(a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2
*x^2 + 4*a*d^2*e^3*f*x + a*d^2*e^4 + (a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*
x + a*d^2*e^4)*cos(d*x + c)^2 + (a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a
*d^2*e^4)*sin(d*x + c)^2 - 2*(a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a*d^
2*e^4)*cos(d*x + c)), x) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 +
3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(5*d*x + 5*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*
a*d^2*e^2*f*x + a*d^2*e^3)*cos(4*d*x + 4*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2
*e^3)*cos(3*d*x + 3*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c
)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*...
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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(csc(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
integral(csc(d*x + c)^3/(a*f^2*x^2 + 2*a*f*x*e + a*e^2 + (a*f^2*x^2 + 2*a*f*x*e + a*e^2)*sin(d*x + c)), x)
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\csc ^{3}{\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(d*x+c)**3/(f*x+e)**2/(a+a*sin(d*x+c)),x)
[Out]
Integral(csc(c + d*x)**3/(e**2*sin(c + d*x) + e**2 + 2*e*f*x*sin(c + d*x) + 2*e*f*x + f**2*x**2*sin(c + d*x) +
f**2*x**2), x)/a
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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(csc(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
Timed out
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{{\sin \left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2\,\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/(sin(c + d*x)^3*(e + f*x)^2*(a + a*sin(c + d*x))),x)
[Out]
int(1/(sin(c + d*x)^3*(e + f*x)^2*(a + a*sin(c + d*x))), x)
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