Integrand size = 19, antiderivative size = 81 \[ \int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \log (\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {3 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^4(c+d x)}{4 d} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2786, 45} \[ \int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \sin ^4(c+d x)}{4 d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {3 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \log (\sin (c+d x))}{d} \]
[In]
[Out]
Rule 45
Rule 2786
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (4 a^3+\frac {a^4}{x}+6 a^2 x+4 a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^4 \log (\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {3 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*}
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \log (\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {3 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^4(c+d x)}{4 d} \]
[In]
[Out]
Time = 1.73 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(-\frac {a^{4} \left (-\frac {1}{4 \csc \left (d x +c \right )^{4}}+\ln \left (\csc \left (d x +c \right )\right )-\frac {4}{\csc \left (d x +c \right )}-\frac {3}{\csc \left (d x +c \right )^{2}}-\frac {4}{3 \csc \left (d x +c \right )^{3}}\right )}{d}\) | \(57\) |
default | \(-\frac {a^{4} \left (-\frac {1}{4 \csc \left (d x +c \right )^{4}}+\ln \left (\csc \left (d x +c \right )\right )-\frac {4}{\csc \left (d x +c \right )}-\frac {3}{\csc \left (d x +c \right )^{2}}-\frac {4}{3 \csc \left (d x +c \right )^{3}}\right )}{d}\) | \(57\) |
risch | \(-i a^{4} x -\frac {13 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}-\frac {13 a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}-\frac {2 i a^{4} c}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {5 a^{4} \sin \left (d x +c \right )}{d}+\frac {a^{4} \cos \left (4 d x +4 c \right )}{32 d}-\frac {a^{4} \sin \left (3 d x +3 c \right )}{3 d}\) | \(120\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \cos \left (d x + c\right )^{4} - 42 \, a^{4} \cos \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 16 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 4 \, a^{4}\right )} \sin \left (d x + c\right )}{12 \, d} \]
[In]
[Out]
\[ \int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=a^{4} \left (\int \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int 4 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx\right ) \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84 \[ \int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 36 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + 48 \, a^{4} \sin \left (d x + c\right )}{12 \, d} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85 \[ \int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 36 \, a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 48 \, a^{4} \sin \left (d x + c\right )}{12 \, d} \]
[In]
[Out]
Time = 9.31 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.46 \[ \int \cot (c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {16\,a^4\,\sin \left (c+d\,x\right )}{3\,d}-\frac {a^4\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a^4\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {7\,a^4\,{\cos \left (c+d\,x\right )}^2}{2\,d}+\frac {a^4\,{\cos \left (c+d\,x\right )}^4}{4\,d}-\frac {4\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d} \]
[In]
[Out]