Optimal. Leaf size=37 \[ \frac {\csc ^3(c+d x)}{3 a d}-\frac {\csc ^4(c+d x)}{4 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 43} \[ \frac {\csc ^3(c+d x)}{3 a d}-\frac {\csc ^4(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^5 (a-x)}{x^5} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {a-x}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (\frac {a}{x^5}-\frac {1}{x^4}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\csc ^3(c+d x)}{3 a d}-\frac {\csc ^4(c+d x)}{4 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 28, normalized size = 0.76 \[ \frac {(4 \sin (c+d x)-3) \csc ^4(c+d x)}{12 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 41, normalized size = 1.11 \[ \frac {4 \, \sin \left (d x + c\right ) - 3}{12 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 26, normalized size = 0.70 \[ \frac {4 \, \sin \left (d x + c\right ) - 3}{12 \, a d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.21, size = 29, normalized size = 0.78 \[ \frac {-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 26, normalized size = 0.70 \[ \frac {4 \, \sin \left (d x + c\right ) - 3}{12 \, a d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 8.64, size = 25, normalized size = 0.68 \[ \frac {\frac {\sin \left (c+d\,x\right )}{3}-\frac {1}{4}}{a\,d\,{\sin \left (c+d\,x\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________