Integrand size = 27, antiderivative size = 75 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {3 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {b \cot (c+d x)}{d}+\frac {3 a \sec (c+d x)}{2 d}-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {b \tan (c+d x)}{d} \]
-3/2*a*arctanh(cos(d*x+c))/d-b*cot(d*x+c)/d+3/2*a*sec(d*x+c)/d-1/2*a*csc(d *x+c)^2*sec(d*x+c)/d+b*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(75)=150\).
Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.29 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {2 b \cot (2 (c+d x))}{d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
(-2*b*Cot[2*(c + d*x)])/d - (a*Csc[(c + d*x)/2]^2)/(8*d) - (3*a*Log[Cos[(c + d*x)/2]])/(2*d) + (3*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/ 2]^2)/(8*d) + (a*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2] )) - (a*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 0.42 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 3317, 3042, 3100, 244, 2009, 3102, 252, 262, 219}
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.
\(\displaystyle \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+b \sin (c+d x)}{\sin (c+d x)^3 \cos (c+d x)^2}dx\) |
\(\Big \downarrow \) 3317 |
\(\displaystyle a \int \csc ^3(c+d x) \sec ^2(c+d x)dx+b \int \csc ^2(c+d x) \sec ^2(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \csc (c+d x)^3 \sec (c+d x)^2dx+b \int \csc (c+d x)^2 \sec (c+d x)^2dx\) |
\(\Big \downarrow \) 3100 |
\(\displaystyle a \int \csc (c+d x)^3 \sec (c+d x)^2dx+\frac {b \int \cot ^2(c+d x) \left (\tan ^2(c+d x)+1\right )d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle a \int \csc (c+d x)^3 \sec (c+d x)^2dx+\frac {b \int \left (\cot ^2(c+d x)+1\right )d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int \csc (c+d x)^3 \sec (c+d x)^2dx+\frac {b (\tan (c+d x)-\cot (c+d x))}{d}\) |
\(\Big \downarrow \) 3102 |
\(\displaystyle \frac {a \int \frac {\sec ^4(c+d x)}{\left (1-\sec ^2(c+d x)\right )^2}d\sec (c+d x)}{d}+\frac {b (\tan (c+d x)-\cot (c+d x))}{d}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {a \left (\frac {\sec ^3(c+d x)}{2 \left (1-\sec ^2(c+d x)\right )}-\frac {3}{2} \int \frac {\sec ^2(c+d x)}{1-\sec ^2(c+d x)}d\sec (c+d x)\right )}{d}+\frac {b (\tan (c+d x)-\cot (c+d x))}{d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {a \left (\frac {\sec ^3(c+d x)}{2 \left (1-\sec ^2(c+d x)\right )}-\frac {3}{2} \left (\int \frac {1}{1-\sec ^2(c+d x)}d\sec (c+d x)-\sec (c+d x)\right )\right )}{d}+\frac {b (\tan (c+d x)-\cot (c+d x))}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a \left (\frac {\sec ^3(c+d x)}{2 \left (1-\sec ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\sec (c+d x))-\sec (c+d x))\right )}{d}+\frac {b (\tan (c+d x)-\cot (c+d x))}{d}\) |
(a*((-3*(ArcTanh[Sec[c + d*x]] - Sec[c + d*x]))/2 + Sec[c + d*x]^3/(2*(1 - Sec[c + d*x]^2))))/d + (b*(-Cot[c + d*x] + Tan[c + d*x]))/d
3.15.49.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[1/f Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]] , x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_S ymbol] :> Simp[1/(f*a^n) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/ 2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1 )/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 0.65 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+b \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )}{d}\) | \(83\) |
default | \(\frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+b \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )}{d}\) | \(83\) |
parallelrisch | \(\frac {\left (12 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -12 a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-24 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-18 a}{8 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 d}\) | \(119\) |
risch | \(-\frac {i \left (3 i a \,{\mathrm e}^{5 i \left (d x +c \right )}-2 i a \,{\mathrm e}^{3 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{i \left (d x +c \right )}+4 b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\) | \(125\) |
norman | \(\frac {\frac {a}{8 d}-\frac {9 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {5 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {5 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(184\) |
1/d*(a*(-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)))+b*(1/sin(d*x+c)/cos(d*x+c)-2*cot(d*x+c)))
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.71 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {6 \, a \cos \left (d x + c\right )^{2} - 3 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 4 \, {\left (2 \, b \cos \left (d x + c\right )^{2} - b\right )} \sin \left (d x + c\right ) - 4 \, a}{4 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )}} \]
1/4*(6*a*cos(d*x + c)^2 - 3*(a*cos(d*x + c)^3 - a*cos(d*x + c))*log(1/2*co s(d*x + c) + 1/2) + 3*(a*cos(d*x + c)^3 - a*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 4*(2*b*cos(d*x + c)^2 - b)*sin(d*x + c) - 4*a)/(d*cos(d*x + c)^3 - d*cos(d*x + c))
Timed out. \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, b {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )}}{4 \, d} \]
1/4*(a*(2*(3*cos(d*x + c)^2 - 2)/(cos(d*x + c)^3 - cos(d*x + c)) - 3*log(c os(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 4*b*(1/tan(d*x + c) - tan(d* x + c)))/d
Time = 0.44 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.55 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {16 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {18 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
1/8*(a*tan(1/2*d*x + 1/2*c)^2 + 12*a*log(abs(tan(1/2*d*x + 1/2*c))) + 4*b* tan(1/2*d*x + 1/2*c) - 16*(b*tan(1/2*d*x + 1/2*c) + a)/(tan(1/2*d*x + 1/2* c)^2 - 1) - (18*a*tan(1/2*d*x + 1/2*c)^2 + 4*b*tan(1/2*d*x + 1/2*c) + a)/t an(1/2*d*x + 1/2*c)^2)/d
Time = 11.38 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.69 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx=\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {-10\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d} \]