Optimal. Leaf size=202 \[ -a^2 x+\frac {5 b^2 x}{2}-\frac {15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}+\frac {15 a b \cos (c+d x)}{4 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {5 b^2 \cot (c+d x)}{2 d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {5 b^2 \cot ^3(c+d x)}{6 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2801, 2671,
294, 308, 209, 2672, 327, 212, 3554, 8} \begin {gather*} -\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}-a^2 x+\frac {15 a b \cos (c+d x)}{4 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac {15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac {5 b^2 \cot ^3(c+d x)}{6 d}+\frac {5 b^2 \cot (c+d x)}{2 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac {5 b^2 x}{2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 209
Rule 212
Rule 294
Rule 308
Rule 327
Rule 2671
Rule 2672
Rule 2801
Rule 3554
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx &=\int \left (b^2 \cos ^2(c+d x) \cot ^4(c+d x)+2 a b \cos (c+d x) \cot ^5(c+d x)+a^2 \cot ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \, dx+(2 a b) \int \cos (c+d x) \cot ^5(c+d x) \, dx+b^2 \int \cos ^2(c+d x) \cot ^4(c+d x) \, dx\\ &=-\frac {a^2 \cot ^5(c+d x)}{5 d}-a^2 \int \cot ^4(c+d x) \, dx-\frac {(2 a b) \text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{d}-\frac {b^2 \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+a^2 \int \cot ^2(c+d x) \, dx+\frac {(5 a b) \text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{2 d}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac {a^2 \cot (c+d x)}{d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-a^2 \int 1 \, dx-\frac {(15 a b) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{4 d}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-a^2 x+\frac {15 a b \cos (c+d x)}{4 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {5 b^2 \cot (c+d x)}{2 d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {5 b^2 \cot ^3(c+d x)}{6 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {(15 a b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{4 d}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-a^2 x+\frac {5 b^2 x}{2}-\frac {15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}+\frac {15 a b \cos (c+d x)}{4 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {5 b^2 \cot (c+d x)}{2 d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {5 b^2 \cot ^3(c+d x)}{6 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.75, size = 351, normalized size = 1.74 \begin {gather*} \frac {-480 a^2 c+1200 b^2 c-480 a^2 d x+1200 b^2 d x+960 a b \cos (c+d x)+\left (-368 a^2+560 b^2\right ) \cot \left (\frac {1}{2} (c+d x)\right )+270 a b \csc ^2\left (\frac {1}{2} (c+d x)\right )-15 a b \csc ^4\left (\frac {1}{2} (c+d x)\right )-1800 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1800 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-270 a b \sec ^2\left (\frac {1}{2} (c+d x)\right )+15 a b \sec ^4\left (\frac {1}{2} (c+d x)\right )-328 a^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+160 b^2 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+96 a^2 \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+\frac {41}{2} a^2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-10 b^2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-\frac {3}{2} a^2 \csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+120 b^2 \sin (2 (c+d x))+368 a^2 \tan \left (\frac {1}{2} (c+d x)\right )-560 b^2 \tan \left (\frac {1}{2} (c+d x)\right )}{480 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.25, size = 216, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a b \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(216\) |
default | \(\frac {a^{2} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a b \left (-\frac {\cos ^{7}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{8}+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(216\) |
risch | \(-a^{2} x +\frac {5 b^{2} x}{2}-\frac {i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {a b \,{\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {a b \,{\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i b^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {180 i a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-180 i b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+135 a \,{\mathrm e}^{9 i \left (d x +c \right )} b -360 i a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+600 i b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-150 a b \,{\mathrm e}^{7 i \left (d x +c \right )}+560 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-800 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-280 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+520 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+150 a b \,{\mathrm e}^{3 i \left (d x +c \right )}+92 i a^{2}-140 i b^{2}-135 a \,{\mathrm e}^{i \left (d x +c \right )} b}{30 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {15 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 d}-\frac {15 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 d}\) | \(321\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 183, normalized size = 0.91 \begin {gather*} -\frac {8 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 20 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} b^{2} + 15 \, a b {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{120 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 306, normalized size = 1.51 \begin {gather*} -\frac {60 \, b^{2} \cos \left (d x + c\right )^{7} + 92 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 140 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 225 \, {\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 225 \, {\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 60 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right ) + 30 \, {\left (2 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} d x \cos \left (d x + c\right )^{4} - 8 \, a b \cos \left (d x + c\right )^{5} - 4 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} d x \cos \left (d x + c\right )^{2} + 25 \, a b \cos \left (d x + c\right )^{3} + 2 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} d x - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sin {\left (c + d x \right )}\right )^{2} \cot ^{6}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 11.45, size = 337, normalized size = 1.67 \begin {gather*} \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1800 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 240 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} {\left (d x + c\right )} - \frac {480 \, {\left (b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {4110 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 11.28, size = 888, normalized size = 4.40 \begin {gather*} \frac {\frac {95\,b^2\,\cos \left (c+d\,x\right )}{384}-\frac {5\,a^2\,\cos \left (c+d\,x\right )}{24}+\frac {5\,a^2\,\cos \left (3\,c+3\,d\,x\right )}{48}-\frac {23\,a^2\,\cos \left (5\,c+5\,d\,x\right )}{240}-\frac {163\,b^2\,\cos \left (3\,c+3\,d\,x\right )}{384}+\frac {71\,b^2\,\cos \left (5\,c+5\,d\,x\right )}{384}-\frac {b^2\,\cos \left (7\,c+7\,d\,x\right )}{128}+\frac {5\,a^2\,\mathrm {atan}\left (\frac {-4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+10\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-10\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (3\,c+3\,d\,x\right )}{8}-\frac {a^2\,\mathrm {atan}\left (\frac {-4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+10\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-10\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (5\,c+5\,d\,x\right )}{8}-\frac {25\,b^2\,\mathrm {atan}\left (\frac {-4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+10\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-10\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (3\,c+3\,d\,x\right )}{16}+\frac {5\,b^2\,\mathrm {atan}\left (\frac {-4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+10\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-10\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (5\,c+5\,d\,x\right )}{16}+\frac {5\,a\,b\,\sin \left (c+d\,x\right )}{4}-\frac {5\,a^2\,\mathrm {atan}\left (\frac {-4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+10\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-10\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (c+d\,x\right )}{4}+\frac {25\,b^2\,\mathrm {atan}\left (\frac {-4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+10\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-10\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (c+d\,x\right )}{8}+\frac {5\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{8}-\frac {5\,a\,b\,\sin \left (3\,c+3\,d\,x\right )}{8}-\frac {17\,a\,b\,\sin \left (4\,c+4\,d\,x\right )}{32}+\frac {a\,b\,\sin \left (5\,c+5\,d\,x\right )}{8}+\frac {a\,b\,\sin \left (6\,c+6\,d\,x\right )}{16}+\frac {75\,a\,b\,\sin \left (c+d\,x\right )\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{32}-\frac {75\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (3\,c+3\,d\,x\right )}{64}+\frac {15\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (5\,c+5\,d\,x\right )}{64}}{d\,{\sin \left (c+d\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________