Integrand size = 29, antiderivative size = 164 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {9 a^2 b \text {arctanh}(\cos (c+d x))}{2 d}-\frac {b^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {2 a^3 \cot (c+d x)}{d}-\frac {3 a b^2 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {9 a^2 b \sec (c+d x)}{2 d}+\frac {b^3 \sec (c+d x)}{d}-\frac {3 a^2 b \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d} \]
-9/2*a^2*b*arctanh(cos(d*x+c))/d-b^3*arctanh(cos(d*x+c))/d-2*a^3*cot(d*x+c )/d-3*a*b^2*cot(d*x+c)/d-1/3*a^3*cot(d*x+c)^3/d+9/2*a^2*b*sec(d*x+c)/d+b^3 *sec(d*x+c)/d-3/2*a^2*b*csc(d*x+c)^2*sec(d*x+c)/d+a^3*tan(d*x+c)/d+3*a*b^2 *tan(d*x+c)/d
Time = 1.71 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.75 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {\csc ^5\left (\frac {1}{2} (c+d x)\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (-8 \left (4 a^3+9 a b^2\right ) \cos (2 (c+d x))+4 \left (4 a^3+9 a b^2\right ) \cos (4 (c+d x))+3 b \left (12 a b+6 \left (5 a^2+2 b^2\right ) \sin (c+d x)-2 \left (9 a^2+2 b^2\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (2 (c+d x))-18 a^2 \sin (3 (c+d x))-4 b^2 \sin (3 (c+d x))+9 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))+2 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))-9 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))-2 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 (c+d x))\right )\right )}{384 d \left (-1+\cot ^2\left (\frac {1}{2} (c+d x)\right )\right )} \]
(Csc[(c + d*x)/2]^5*Sec[(c + d*x)/2]^3*(-8*(4*a^3 + 9*a*b^2)*Cos[2*(c + d* x)] + 4*(4*a^3 + 9*a*b^2)*Cos[4*(c + d*x)] + 3*b*(12*a*b + 6*(5*a^2 + 2*b^ 2)*Sin[c + d*x] - 2*(9*a^2 + 2*b^2)*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[2*(c + d*x)] - 18*a^2*Sin[3*(c + d*x)] - 4*b^2*Sin[3*(c + d* x)] + 9*a^2*Log[Cos[(c + d*x)/2]]*Sin[4*(c + d*x)] + 2*b^2*Log[Cos[(c + d* x)/2]]*Sin[4*(c + d*x)] - 9*a^2*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] - 2 *b^2*Log[Sin[(c + d*x)/2]]*Sin[4*(c + d*x)])))/(384*d*(-1 + Cot[(c + d*x)/ 2]^2))
Time = 0.48 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3391, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^3}{\sin (c+d x)^4 \cos (c+d x)^2}dx\) |
| \(\Big \downarrow \) 3391 |
| \(\displaystyle \int \left (a^3 \csc ^4(c+d x) \sec ^2(c+d x)+3 a^2 b \csc ^3(c+d x) \sec ^2(c+d x)+3 a b^2 \csc ^2(c+d x) \sec ^2(c+d x)+b^3 \csc (c+d x) \sec ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \tan (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {2 a^3 \cot (c+d x)}{d}-\frac {9 a^2 b \text {arctanh}(\cos (c+d x))}{2 d}+\frac {9 a^2 b \sec (c+d x)}{2 d}-\frac {3 a^2 b \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {3 a b^2 \cot (c+d x)}{d}-\frac {b^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {b^3 \sec (c+d x)}{d}\) |
(-9*a^2*b*ArcTanh[Cos[c + d*x]])/(2*d) - (b^3*ArcTanh[Cos[c + d*x]])/d - ( 2*a^3*Cot[c + d*x])/d - (3*a*b^2*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3 *d) + (9*a^2*b*Sec[c + d*x])/(2*d) + (b^3*Sec[c + d*x])/d - (3*a^2*b*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (a^3*Tan[c + d*x])/d + (3*a*b^2*Tan[c + d* x])/d
3.15.63.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m] && (G tQ[m, 0] || IntegerQ[n])
Time = 0.85 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+3 a^{2} b \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a \,b^{2} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+b^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(169\) |
| default | \(\frac {a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+3 a^{2} b \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a \,b^{2} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+b^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(169\) |
| parallelrisch | \(\frac {108 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (a^{2}+\frac {2 b^{2}}{9}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+9 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +\left (20 a^{3}+36 a \,b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-90 a^{3}-216 a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+9 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +\left (20 a^{3}+36 a \,b^{2}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-162 a^{2} b -48 b^{3}}{24 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 d}\) | \(210\) |
| risch | \(\frac {27 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-36 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-45 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-18 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+32 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+72 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+45 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+18 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-16 i a^{3}-36 i a \,b^{2}-27 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-6 b^{3} {\mathrm e}^{i \left (d x +c \right )}}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {9 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {9 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}\) | \(291\) |
| norman | \(\frac {\frac {a^{3}}{24 d}+\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {\left (9 a^{2} b +4 b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {\left (63 a^{2} b +16 b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {\left (135 a^{2} b +48 b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {9 a \left (a^{2}+4 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {9 a \left (a^{2}+4 b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {21 a \left (3 a^{2}+8 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {21 a \left (3 a^{2}+8 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a \left (23 a^{2}+36 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a \left (23 a^{2}+36 b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {3 a^{2} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {3 a^{2} b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {3 b \left (13 a^{2}+4 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {b \left (9 a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(413\) |
1/d*(a^3*(-1/3/sin(d*x+c)^3/cos(d*x+c)+4/3/sin(d*x+c)/cos(d*x+c)-8/3*cot(d *x+c))+3*a^2*b*(-1/2/sin(d*x+c)^2/cos(d*x+c)+3/2/cos(d*x+c)+3/2*ln(csc(d*x +c)-cot(d*x+c)))+3*a*b^2*(1/sin(d*x+c)/cos(d*x+c)-2*cot(d*x+c))+b^3*(1/cos (d*x+c)+ln(csc(d*x+c)-cot(d*x+c))))
Time = 0.28 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.54 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {8 \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + 12 \, a^{3} + 36 \, a b^{2} - 12 \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 6 \, {\left (6 \, a^{2} b + 2 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \]
-1/12*(8*(4*a^3 + 9*a*b^2)*cos(d*x + c)^4 + 12*a^3 + 36*a*b^2 - 12*(4*a^3 + 9*a*b^2)*cos(d*x + c)^2 + 3*((9*a^2*b + 2*b^3)*cos(d*x + c)^3 - (9*a^2*b + 2*b^3)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*((9*a ^2*b + 2*b^3)*cos(d*x + c)^3 - (9*a^2*b + 2*b^3)*cos(d*x + c))*log(-1/2*co s(d*x + c) + 1/2)*sin(d*x + c) + 6*(6*a^2*b + 2*b^3 - (9*a^2*b + 2*b^3)*co s(d*x + c)^2)*sin(d*x + c))/((d*cos(d*x + c)^3 - d*cos(d*x + c))*sin(d*x + c))
Timed out. \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.99 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {9 \, a^{2} b {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 6 \, b^{3} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 36 \, a b^{2} {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} - 4 \, a^{3} {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )}}{12 \, d} \]
1/12*(9*a^2*b*(2*(3*cos(d*x + c)^2 - 2)/(cos(d*x + c)^3 - cos(d*x + c)) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) + 6*b^3*(2/cos(d*x + c) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) - 36*a*b^2*(1/tan(d*x + c) - tan(d*x + c)) - 4*a^3*((6*tan(d*x + c)^2 + 1)/tan(d*x + c)^3 - 3*tan( d*x + c)))/d
Time = 0.52 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.49 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 21 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {48 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2} b + b^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {198 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 44 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
1/24*(a^3*tan(1/2*d*x + 1/2*c)^3 + 9*a^2*b*tan(1/2*d*x + 1/2*c)^2 + 21*a^3 *tan(1/2*d*x + 1/2*c) + 36*a*b^2*tan(1/2*d*x + 1/2*c) + 12*(9*a^2*b + 2*b^ 3)*log(abs(tan(1/2*d*x + 1/2*c))) - 48*(a^3*tan(1/2*d*x + 1/2*c) + 3*a*b^2 *tan(1/2*d*x + 1/2*c) + 3*a^2*b + b^3)/(tan(1/2*d*x + 1/2*c)^2 - 1) - (198 *a^2*b*tan(1/2*d*x + 1/2*c)^3 + 44*b^3*tan(1/2*d*x + 1/2*c)^3 + 21*a^3*tan (1/2*d*x + 1/2*c)^2 + 36*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 9*a^2*b*tan(1/2*d* x + 1/2*c) + a^3)/tan(1/2*d*x + 1/2*c)^3)/d
Time = 11.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.33 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {9\,a^2\,b}{2}+b^3\right )}{d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {20\,a^3}{3}+12\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (23\,a^3+60\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (51\,a^2\,b+16\,b^3\right )+\frac {a^3}{3}+3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {7\,a^3}{8}+\frac {3\,a\,b^2}{2}\right )}{d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d} \]
(log(tan(c/2 + (d*x)/2))*((9*a^2*b)/2 + b^3))/d + (a^3*tan(c/2 + (d*x)/2)^ 3)/(24*d) - (tan(c/2 + (d*x)/2)^2*(12*a*b^2 + (20*a^3)/3) - tan(c/2 + (d*x )/2)^4*(60*a*b^2 + 23*a^3) - tan(c/2 + (d*x)/2)^3*(51*a^2*b + 16*b^3) + a^ 3/3 + 3*a^2*b*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^3 - 8*tan(c/2 + (d*x)/2)^5)) + (tan(c/2 + (d*x)/2)*((3*a*b^2)/2 + (7*a^3)/8))/d + (3*a^2* b*tan(c/2 + (d*x)/2)^2)/(8*d)