Optimal. Leaf size=148 \[ \frac {a^4 \sin ^4(c+d x)}{4 d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^4 \sin ^2(c+d x)}{d}-\frac {4 a^4 \sin (c+d x)}{d}-\frac {a^4 \csc ^4(c+d x)}{4 d}-\frac {4 a^4 \csc ^3(c+d x)}{3 d}-\frac {2 a^4 \csc ^2(c+d x)}{d}+\frac {4 a^4 \csc (c+d x)}{d}-\frac {10 a^4 \log (\sin (c+d x))}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2707, 88} \[ \frac {a^4 \sin ^4(c+d x)}{4 d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^4 \sin ^2(c+d x)}{d}-\frac {4 a^4 \sin (c+d x)}{d}-\frac {a^4 \csc ^4(c+d x)}{4 d}-\frac {4 a^4 \csc ^3(c+d x)}{3 d}-\frac {2 a^4 \csc ^2(c+d x)}{d}+\frac {4 a^4 \csc (c+d x)}{d}-\frac {10 a^4 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 2707
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^6}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-4 a^3+\frac {a^8}{x^5}+\frac {4 a^7}{x^4}+\frac {4 a^6}{x^3}-\frac {4 a^5}{x^2}-\frac {10 a^4}{x}+4 a^2 x+4 a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {4 a^4 \csc (c+d x)}{d}-\frac {2 a^4 \csc ^2(c+d x)}{d}-\frac {4 a^4 \csc ^3(c+d x)}{3 d}-\frac {a^4 \csc ^4(c+d x)}{4 d}-\frac {10 a^4 \log (\sin (c+d x))}{d}-\frac {4 a^4 \sin (c+d x)}{d}+\frac {2 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^4(c+d x)}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 96, normalized size = 0.65 \[ \frac {a^4 \left (3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+24 \sin ^2(c+d x)-48 \sin (c+d x)-3 \csc ^4(c+d x)-16 \csc ^3(c+d x)-24 \csc ^2(c+d x)+48 \csc (c+d x)-120 \log (\sin (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.97, size = 144, normalized size = 0.97 \[ \frac {24 \, a^{4} \cos \left (d x + c\right )^{8} - 128 \, a^{4} \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) - 288 \, a^{4} \cos \left (d x + c\right )^{6} + 615 \, a^{4} \cos \left (d x + c\right )^{4} - 270 \, a^{4} \cos \left (d x + c\right )^{2} - 105 \, a^{4} - 960 \, {\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{96 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.46, size = 134, normalized size = 0.91 \[ \frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 24 \, a^{4} \sin \left (d x + c\right )^{2} - 120 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 48 \, a^{4} \sin \left (d x + c\right ) + \frac {250 \, a^{4} \sin \left (d x + c\right )^{4} + 48 \, a^{4} \sin \left (d x + c\right )^{3} - 24 \, a^{4} \sin \left (d x + c\right )^{2} - 16 \, a^{4} \sin \left (d x + c\right ) - 3 \, a^{4}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.44, size = 129, normalized size = 0.87 \[ -\frac {11 a^{4} \left (\cos ^{4}\left (d x +c \right )\right )}{4 d}-\frac {11 a^{4} \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}-\frac {10 a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {3 a^{4} \left (\cos ^{6}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{2}}-\frac {4 a^{4} \left (\cos ^{6}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}-\frac {a^{4} \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{4} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.72, size = 120, normalized size = 0.81 \[ \frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 24 \, a^{4} \sin \left (d x + c\right )^{2} - 120 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) - 48 \, a^{4} \sin \left (d x + c\right ) + \frac {48 \, a^{4} \sin \left (d x + c\right )^{3} - 24 \, a^{4} \sin \left (d x + c\right )^{2} - 16 \, a^{4} \sin \left (d x + c\right ) - 3 \, a^{4}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.07, size = 368, normalized size = 2.49 \[ \frac {3\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6\,d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {10\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {104\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-119\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+120\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-\frac {1135\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4}+80\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-73\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+48\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {75\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-\frac {40\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+10\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {8\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {a^4}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {9\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}+\frac {10\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
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]
________________________________________________________________________________________