Integrand size = 23, antiderivative size = 385 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {b^{5/3} \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^2-b^2\right )^2 d}-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \left (a^2-b^2\right )^2 d}+\frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))} \] Output:
-1/3*b^(5/3)*(2*a^2-3*a^(4/3)*b^(2/3)+b^2)*arctan(1/3*(a^(1/3)-2*b^(1/3)*s in(d*x+c))*3^(1/2)/a^(1/3))*3^(1/2)/a^(2/3)/(a^2-b^2)^2/d-1/4*(a+4*b)*ln(1 -sin(d*x+c))/(a+b)^2/d+1/4*(a-4*b)*ln(1+sin(d*x+c))/(a-b)^2/d+1/3*b^(5/3)* (2*a^2+3*a^(4/3)*b^(2/3)+b^2)*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(2/3)/(a^2- b^2)^2/d-1/6*b^(5/3)*(2*a^2+3*a^(4/3)*b^(2/3)+b^2)*ln(a^(2/3)-a^(1/3)*b^(1 /3)*sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(2/3)/(a^2-b^2)^2/d+1/3*b*(a^2+2*b^ 2)*ln(a+b*sin(d*x+c)^3)/(a^2-b^2)^2/d+1/4/(a+b)/d/(1-sin(d*x+c))-1/4/(a-b) /d/(1+sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.58 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.94 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\frac {4 \sqrt {3} b^{5/3} \left (2 a^2+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3} \left (a^2-b^2\right )^2}+\frac {3 (a+4 b) \log (1-\sin (c+d x))}{(a+b)^2}-\frac {3 (a-4 b) \log (1+\sin (c+d x))}{(a-b)^2}-\frac {4 b^{5/3} \left (2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{2/3} \left (a^2-b^2\right )^2}+\frac {2 b^{5/3} \left (2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{2/3} \left (a^2-b^2\right )^2}-\frac {4 b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac {3}{(a+b) (-1+\sin (c+d x))}+\frac {18 b^3 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{\left (a^2-b^2\right )^2}+\frac {3}{(a-b) (1+\sin (c+d x))}}{12 d} \] Input:
Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]
Output:
-1/12*((4*Sqrt[3]*b^(5/3)*(2*a^2 + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(a^(2/3)*(a^2 - b^2)^2) + (3*(a + 4*b)*Log[1 - Sin[c + d*x]])/(a + b)^2 - (3*(a - 4*b)*Log[1 + Sin[c + d*x]])/(a - b)^2 - (4*b^(5/3)*(2*a^2 + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(2/3)*( a^2 - b^2)^2) + (2*b^(5/3)*(2*a^2 + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin [c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(a^(2/3)*(a^2 - b^2)^2) - (4*b*(a^2 + 2*b^2)*Log[a + b*Sin[c + d*x]^3])/(a^2 - b^2)^2 + 3/((a + b)*(-1 + Sin[c + d*x])) + (18*b^3*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)] *Sin[c + d*x]^2)/(a^2 - b^2)^2 + 3/((a - b)*(1 + Sin[c + d*x])))/d
Time = 0.74 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3702, 7276, 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.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^3 \left (a+b \sin (c+d x)^3\right )}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(c+d x)\right )^2 \left (b \sin ^3(c+d x)+a\right )}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\frac {\left (2 a^2-3 b \sin (c+d x) a+b^2+\left (a^2+2 b^2\right ) \sin ^2(c+d x)\right ) b^2}{\left (a^2-b^2\right )^2 \left (b \sin ^3(c+d x)+a\right )}+\frac {-a-4 b}{4 (a+b)^2 (\sin (c+d x)-1)}+\frac {a-4 b}{4 (a-b)^2 (\sin (c+d x)+1)}+\frac {1}{4 (a+b) (\sin (c+d x)-1)^2}+\frac {1}{4 (a-b) (\sin (c+d x)+1)^2}\right )d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \left (a^2-b^2\right )^2}+\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2}+\frac {1}{4 (a+b) (1-\sin (c+d x))}-\frac {1}{4 (a-b) (\sin (c+d x)+1)}-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2}+\frac {(a-4 b) \log (\sin (c+d x)+1)}{4 (a-b)^2}}{d}\) |
Input:
Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3),x]
Output:
(-((b^(5/3)*(2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)* Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a^2 - b^2)^2)) - ((a + 4*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2) + ((a - 4*b)*Log[1 + Sin[c + d* x]])/(4*(a - b)^2) + (b^(5/3)*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(1/3 ) + b^(1/3)*Sin[c + d*x]])/(3*a^(2/3)*(a^2 - b^2)^2) - (b^(5/3)*(2*a^2 + 3 *a^(4/3)*b^(2/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/ 3)*Sin[c + d*x]^2])/(6*a^(2/3)*(a^2 - b^2)^2) + (b*(a^2 + 2*b^2)*Log[a + b *Sin[c + d*x]^3])/(3*(a^2 - b^2)^2) + 1/(4*(a + b)*(1 - Sin[c + d*x])) - 1 /(4*(a - b)*(1 + Sin[c + d*x])))/d
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x _)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 1.90 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{\left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-4 b -a \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}-\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -4 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}+\frac {\left (\left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-3 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (a^{2}+2 b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )^{3}\right )}{3 b}\right ) b^{2}}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\) | \(372\) |
| default | \(\frac {-\frac {1}{\left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-4 b -a \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{2}}-\frac {1}{\left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -4 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{2}}+\frac {\left (\left (2 a^{2}+b^{2}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-3 a b \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (a^{2}+2 b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )^{3}\right )}{3 b}\right ) b^{2}}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\) | \(372\) |
| risch | \(-\frac {2 i a^{4} b \,d^{2} c}{a^{6} d^{3}-2 a^{4} b^{2} d^{3}+d^{3} b^{4} a^{2}}-\frac {i a c}{2 d \left (a^{2}-2 a b +b^{2}\right )}-\frac {i a x}{2 \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 i b c}{d \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 i b x}{a^{2}-2 a b +b^{2}}-\frac {2 i a^{4} b \,d^{3} x}{a^{6} d^{3}-2 a^{4} b^{2} d^{3}+d^{3} b^{4} a^{2}}+\frac {2 i b x}{a^{2}+2 a b +b^{2}}+\frac {i a x}{2 a^{2}+4 a b +2 b^{2}}+\frac {2 i b c}{d \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 i d^{2} b^{3} a^{2} c}{a^{6} d^{3}-2 a^{4} b^{2} d^{3}+d^{3} b^{4} a^{2}}+\frac {i a c}{2 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 i d^{3} b^{3} a^{2} x}{a^{6} d^{3}-2 a^{4} b^{2} d^{3}+d^{3} b^{4} a^{2}}+\frac {i a \,{\mathrm e}^{3 i \left (d x +c \right )}-i a \,{\mathrm e}^{i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{d \left (-a^{2}+b^{2}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a}{2 d \left (a^{2}-2 a b +b^{2}\right )}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b}{d \left (a^{2}-2 a b +b^{2}\right )}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b}{d \left (a^{2}+2 a b +b^{2}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a}{2 d \left (a^{2}+2 a b +b^{2}\right )}+8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (13824 a^{6} d^{3}-27648 a^{4} b^{2} d^{3}+13824 d^{3} b^{4} a^{2}\right ) \textit {\_Z}^{3}+\left (-1728 a^{4} b \,d^{2}-3456 d^{2} b^{3} a^{2}\right ) \textit {\_Z}^{2}+72 a^{2} b^{2} d \textit {\_Z} -b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {3456 i a^{7} d^{2}}{8 a^{2} b^{3}+b^{5}}+\frac {6912 i a^{5} b^{2} d^{2}}{8 a^{2} b^{3}+b^{5}}-\frac {3456 i a^{3} b^{4} d^{2}}{8 a^{2} b^{3}+b^{5}}\right ) \textit {\_R}^{2}+\left (\frac {480 i a^{5} b d}{8 a^{2} b^{3}+b^{5}}+\frac {768 i a^{3} b^{3} d}{8 a^{2} b^{3}+b^{5}}+\frac {48 i a \,b^{5} d}{8 a^{2} b^{3}+b^{5}}\right ) \textit {\_R} -\frac {14 i a^{3} b^{2}}{8 a^{2} b^{3}+b^{5}}-\frac {4 i a \,b^{4}}{8 a^{2} b^{3}+b^{5}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {8 a^{2} b^{3}}{8 a^{2} b^{3}+b^{5}}-\frac {b^{5}}{8 a^{2} b^{3}+b^{5}}\right )\right )\) | \(867\) |
Input:
int(sec(d*x+c)^3/(a+b*sin(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/(4*a+4*b)/(sin(d*x+c)-1)+1/4/(a+b)^2*(-4*b-a)*ln(sin(d*x+c)-1)-1/( 4*a-4*b)/(1+sin(d*x+c))+1/4*(a-4*b)/(a-b)^2*ln(1+sin(d*x+c))+((2*a^2+b^2)* (1/3/b/(1/b*a)^(2/3)*ln(sin(d*x+c)+(1/b*a)^(1/3))-1/6/b/(1/b*a)^(2/3)*ln(s in(d*x+c)^2-(1/b*a)^(1/3)*sin(d*x+c)+(1/b*a)^(2/3))+1/3/b/(1/b*a)^(2/3)*3^ (1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))-3*a*b*(-1/3/b/(1 /b*a)^(1/3)*ln(sin(d*x+c)+(1/b*a)^(1/3))+1/6/b/(1/b*a)^(1/3)*ln(sin(d*x+c) ^2-(1/b*a)^(1/3)*sin(d*x+c)+(1/b*a)^(2/3))+1/3*3^(1/2)/b/(1/b*a)^(1/3)*arc tan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))+1/3*(a^2+2*b^2)/b*ln(a+b* sin(d*x+c)^3))*b^2/(a-b)^2/(a+b)^2)
Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 10135, normalized size of antiderivative = 26.32 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \] Input:
integrate(sec(d*x+c)**3/(a+b*sin(d*x+c)**3),x)
Output:
Integral(sec(c + d*x)**3/(a + b*sin(c + d*x)**3), x)
Time = 0.12 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.22 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm="maxima")
Output:
-1/36*(4*sqrt(3)*(a*b^2*(9*(a/b)^(2/3) + 4) - b^3*(3*(a/b)^(1/3) + 4*a/b) - 2*a^2*b*(3*(a/b)^(1/3) + a/b) + 2*a^3)*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2*sin(d*x + c))/(a/b)^(1/3))/((a^4*(a/b)^(2/3) - 2*a^2*b^2*(a/b)^(2/3) + b^4*(a/b)^(2/3))*(a/b)^(1/3)) - 6*(b^3*(4*(a/b)^(2/3) - 1) + 2*a^2*b*((a/ b)^(2/3) - 1) - 3*a*b^2*(a/b)^(1/3))*log(sin(d*x + c)^2 - (a/b)^(1/3)*sin( d*x + c) + (a/b)^(2/3))/(a^4*(a/b)^(2/3) - 2*a^2*b^2*(a/b)^(2/3) + b^4*(a/ b)^(2/3)) - 12*(b^3*(2*(a/b)^(2/3) + 1) + a^2*b*((a/b)^(2/3) + 2) + 3*a*b^ 2*(a/b)^(1/3))*log((a/b)^(1/3) + sin(d*x + c))/(a^4*(a/b)^(2/3) - 2*a^2*b^ 2*(a/b)^(2/3) + b^4*(a/b)^(2/3)) - 9*(a - 4*b)*log(sin(d*x + c) + 1)/(a^2 - 2*a*b + b^2) + 9*(a + 4*b)*log(sin(d*x + c) - 1)/(a^2 + 2*a*b + b^2) + 1 8*(a*sin(d*x + c) - b)/((a^2 - b^2)*sin(d*x + c)^2 - a^2 + b^2))/d
Time = 0.41 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.38 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {{\left (3 \, a^{5} b^{4} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 6 \, a^{3} b^{6} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a b^{8} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{6} b^{3} d + 3 \, a^{4} b^{5} d - b^{9} d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{3 \, {\left (a^{9} b d^{2} - 4 \, a^{7} b^{3} d^{2} + 6 \, a^{5} b^{5} d^{2} - 4 \, a^{3} b^{7} d^{2} + a b^{9} d^{2}\right )}} + \frac {{\left (a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{3 \, {\left (a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac {{\left (a + 4 \, b\right )} \log \left ({\left | -\sin \left (d x + c\right ) + 1 \right |}\right )}{4 \, {\left (a^{2} d + 2 \, a b d + b^{2} d\right )}} + \frac {{\left (a - 4 \, b\right )} \log \left ({\left | -\sin \left (d x + c\right ) - 1 \right |}\right )}{4 \, {\left (a^{2} d - 2 \, a b d + b^{2} d\right )}} + \frac {{\left (3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b + {\left (2 \, a^{2} b + b^{3}\right )} \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (\sqrt {3} a^{5} - 2 \, \sqrt {3} a^{3} b^{2} + \sqrt {3} a b^{4}\right )} d} - \frac {{\left (3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b - {\left (2 \, a^{2} b + b^{3}\right )} \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d} + \frac {a^{2} b - b^{3} - {\left (a^{3} - a b^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left (a + b\right )}^{2} {\left (a - b\right )}^{2} d {\left (\sin \left (d x + c\right ) + 1\right )} {\left (\sin \left (d x + c\right ) - 1\right )}} \] Input:
integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm="giac")
Output:
1/3*(3*a^5*b^4*d*(-a/b)^(1/3) - 6*a^3*b^6*d*(-a/b)^(1/3) + 3*a*b^8*d*(-a/b )^(1/3) - 2*a^6*b^3*d + 3*a^4*b^5*d - b^9*d)*(-a/b)^(1/3)*log(abs(-(-a/b)^ (1/3) + sin(d*x + c)))/(a^9*b*d^2 - 4*a^7*b^3*d^2 + 6*a^5*b^5*d^2 - 4*a^3* b^7*d^2 + a*b^9*d^2) + 1/3*(a^2*b + 2*b^3)*log(abs(b*sin(d*x + c)^3 + a))/ (a^4*d - 2*a^2*b^2*d + b^4*d) - 1/4*(a + 4*b)*log(abs(-sin(d*x + c) + 1))/ (a^2*d + 2*a*b*d + b^2*d) + 1/4*(a - 4*b)*log(abs(-sin(d*x + c) - 1))/(a^2 *d - 2*a*b*d + b^2*d) + (3*(-a*b^2)^(2/3)*a*b + (2*a^2*b + b^3)*(-a*b^2)^( 1/3))*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/((s qrt(3)*a^5 - 2*sqrt(3)*a^3*b^2 + sqrt(3)*a*b^4)*d) - 1/6*(3*(-a*b^2)^(2/3) *a*b - (2*a^2*b + b^3)*(-a*b^2)^(1/3))*log(sin(d*x + c)^2 + (-a/b)^(1/3)*s in(d*x + c) + (-a/b)^(2/3))/((a^5 - 2*a^3*b^2 + a*b^4)*d) + 1/2*(a^2*b - b ^3 - (a^3 - a*b^2)*sin(d*x + c))/((a + b)^2*(a - b)^2*d*(sin(d*x + c) + 1) *(sin(d*x + c) - 1))
Time = 36.25 (sec) , antiderivative size = 898, normalized size of antiderivative = 2.33 \[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\left (\sum _{k=1}^3\ln \left (-\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (-\frac {28\,a\,b^7-6\,a^3\,b^5}{a^4-2\,a^2\,b^2+b^4}+\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (-\frac {18\,a^5\,b^4-\frac {219\,a^3\,b^6}{4}+12\,a\,b^8}{a^4-2\,a^2\,b^2+b^4}+\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (\frac {\frac {27\,a^7\,b^3}{2}-87\,a^5\,b^5+\frac {51\,a^3\,b^7}{2}+48\,a\,b^9}{a^4-2\,a^2\,b^2+b^4}+\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (\frac {180\,a^7\,b^4-324\,a^5\,b^6+108\,a^3\,b^8+36\,a\,b^{10}}{a^4-2\,a^2\,b^2+b^4}+\frac {\sin \left (c+d\,x\right )\,\left (216\,a^8\,b^3+216\,a^6\,b^5-1080\,a^4\,b^7+648\,a^2\,b^9\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (-600\,a^4\,b^6+552\,a^2\,b^8+48\,b^{10}\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (-171\,a^4\,b^5+120\,a^2\,b^7+96\,b^9\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (61\,a^2\,b^6+40\,b^8\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {a\,b^6}{2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,b^7\,\sin \left (c+d\,x\right )}{a^4-2\,a^2\,b^2+b^4}\right )\,\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\right )+\frac {\frac {b}{2\,\left (a^2-b^2\right )}-\frac {a\,\sin \left (c+d\,x\right )}{2\,\left (a^2-b^2\right )}}{{\sin \left (c+d\,x\right )}^2-1}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (a+4\,b\right )}{4\,a^2+8\,a\,b+4\,b^2}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-4\,b\right )}{4\,a^2-8\,a\,b+4\,b^2}}{d} \] Input:
int(1/(cos(c + d*x)^3*(a + b*sin(c + d*x)^3)),x)
Output:
(symsum(log((a*b^6)/(2*(a^4 + b^4 - 2*a^2*b^2)) - root(54*a^4*b^2*z^3 - 27 *a^2*b^4*z^3 - 27*a^6*z^3 + 54*a^2*b^3*z^2 + 27*a^4*b*z^2 - 9*a^2*b^2*z + b^3, z, k)*(root(54*a^4*b^2*z^3 - 27*a^2*b^4*z^3 - 27*a^6*z^3 + 54*a^2*b^3 *z^2 + 27*a^4*b*z^2 - 9*a^2*b^2*z + b^3, z, k)*(root(54*a^4*b^2*z^3 - 27*a ^2*b^4*z^3 - 27*a^6*z^3 + 54*a^2*b^3*z^2 + 27*a^4*b*z^2 - 9*a^2*b^2*z + b^ 3, z, k)*((48*a*b^9 + (51*a^3*b^7)/2 - 87*a^5*b^5 + (27*a^7*b^3)/2)/(a^4 + b^4 - 2*a^2*b^2) + root(54*a^4*b^2*z^3 - 27*a^2*b^4*z^3 - 27*a^6*z^3 + 54 *a^2*b^3*z^2 + 27*a^4*b*z^2 - 9*a^2*b^2*z + b^3, z, k)*((36*a*b^10 + 108*a ^3*b^8 - 324*a^5*b^6 + 180*a^7*b^4)/(a^4 + b^4 - 2*a^2*b^2) + (sin(c + d*x )*(648*a^2*b^9 - 1080*a^4*b^7 + 216*a^6*b^5 + 216*a^8*b^3))/(4*(a^4 + b^4 - 2*a^2*b^2))) + (sin(c + d*x)*(48*b^10 + 552*a^2*b^8 - 600*a^4*b^6))/(4*( a^4 + b^4 - 2*a^2*b^2))) - (12*a*b^8 - (219*a^3*b^6)/4 + 18*a^5*b^4)/(a^4 + b^4 - 2*a^2*b^2) + (sin(c + d*x)*(96*b^9 + 120*a^2*b^7 - 171*a^4*b^5))/( 4*(a^4 + b^4 - 2*a^2*b^2))) - (28*a*b^7 - 6*a^3*b^5)/(a^4 + b^4 - 2*a^2*b^ 2) + (sin(c + d*x)*(40*b^8 + 61*a^2*b^6))/(4*(a^4 + b^4 - 2*a^2*b^2))) + ( 2*b^7*sin(c + d*x))/(a^4 + b^4 - 2*a^2*b^2))*root(54*a^4*b^2*z^3 - 27*a^2* b^4*z^3 - 27*a^6*z^3 + 54*a^2*b^3*z^2 + 27*a^4*b*z^2 - 9*a^2*b^2*z + b^3, z, k), k, 1, 3) + (b/(2*(a^2 - b^2)) - (a*sin(c + d*x))/(2*(a^2 - b^2)))/( sin(c + d*x)^2 - 1) - (log(sin(c + d*x) - 1)*(a + 4*b))/(8*a*b + 4*a^2 + 4 *b^2) + (log(sin(c + d*x) + 1)*(a - 4*b))/(4*a^2 - 8*a*b + 4*b^2))/d
\[ \int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\sec \left (d x +c \right )^{3}}{\sin \left (d x +c \right )^{3} b +a}d x \] Input:
int(sec(d*x+c)^3/(a+b*sin(d*x+c)^3),x)
Output:
int(sec(c + d*x)**3/(sin(c + d*x)**3*b + a),x)