Optimal. Leaf size=109 \[ -\frac {\csc ^{10}(c+d x)}{10 a d}+\frac {\csc ^9(c+d x)}{9 a d}+\frac {\csc ^8(c+d x)}{4 a d}-\frac {2 \csc ^7(c+d x)}{7 a d}-\frac {\csc ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 109, 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, 88} \[ -\frac {\csc ^{10}(c+d x)}{10 a d}+\frac {\csc ^9(c+d x)}{9 a d}+\frac {\csc ^8(c+d x)}{4 a d}-\frac {2 \csc ^7(c+d x)}{7 a d}-\frac {\csc ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cot ^7(c+d x) \csc ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^{11} (a-x)^3 (a+x)^2}{x^{11}} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {a^4 \operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x^{11}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^4 \operatorname {Subst}\left (\int \left (\frac {a^5}{x^{11}}-\frac {a^4}{x^{10}}-\frac {2 a^3}{x^9}+\frac {2 a^2}{x^8}+\frac {a}{x^7}-\frac {1}{x^6}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\csc ^5(c+d x)}{5 a d}-\frac {\csc ^6(c+d x)}{6 a d}-\frac {2 \csc ^7(c+d x)}{7 a d}+\frac {\csc ^8(c+d x)}{4 a d}+\frac {\csc ^9(c+d x)}{9 a d}-\frac {\csc ^{10}(c+d x)}{10 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 68, normalized size = 0.62 \[ \frac {\csc ^5(c+d x) \left (-126 \csc ^5(c+d x)+140 \csc ^4(c+d x)+315 \csc ^3(c+d x)-360 \csc ^2(c+d x)-210 \csc (c+d x)+252\right )}{1260 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 120, normalized size = 1.10 \[ \frac {210 \, \cos \left (d x + c\right )^{4} - 105 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (63 \, \cos \left (d x + c\right )^{4} - 36 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 21}{1260 \, {\left (a d \cos \left (d x + c\right )^{10} - 5 \, a d \cos \left (d x + c\right )^{8} + 10 \, a d \cos \left (d x + c\right )^{6} - 10 \, a d \cos \left (d x + c\right )^{4} + 5 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 66, normalized size = 0.61 \[ \frac {252 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 360 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) - 126}{1260 \, a d \sin \left (d x + c\right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.58, size = 69, normalized size = 0.63 \[ \frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {2}{7 \sin \left (d x +c \right )^{7}}+\frac {1}{9 \sin \left (d x +c \right )^{9}}+\frac {1}{4 \sin \left (d x +c \right )^{8}}-\frac {1}{10 \sin \left (d x +c \right )^{10}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 66, normalized size = 0.61 \[ \frac {252 \, \sin \left (d x + c\right )^{5} - 210 \, \sin \left (d x + c\right )^{4} - 360 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )^{2} + 140 \, \sin \left (d x + c\right ) - 126}{1260 \, a d \sin \left (d x + c\right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.08, size = 66, normalized size = 0.61 \[ \frac {252\,{\sin \left (c+d\,x\right )}^5-210\,{\sin \left (c+d\,x\right )}^4-360\,{\sin \left (c+d\,x\right )}^3+315\,{\sin \left (c+d\,x\right )}^2+140\,\sin \left (c+d\,x\right )-126}{1260\,a\,d\,{\sin \left (c+d\,x\right )}^{10}} \]
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]
________________________________________________________________________________________