Integrand size = 21, antiderivative size = 119 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {17 a^3 x}{2}-\frac {6 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {25 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d} \] Output:
17/2*a^3*x-6*a^3*cos(d*x+c)/d+1/3*a^3*cos(d*x+c)^3/d+2/3*a^3*cos(d*x+c)/d/ (1-sin(d*x+c))^2-25/3*a^3*cos(d*x+c)/d/(1-sin(d*x+c))-3/2*a^3*cos(d*x+c)*s in(d*x+c)/d
Time = 7.80 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.49 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {(a+a \sin (c+d x))^3 \left (102 (c+d x)-69 \cos (c+d x)+\cos (3 (c+d x))+\frac {8}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {16 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {200 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-9 \sin (2 (c+d x))\right )}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]
Output:
((a + a*Sin[c + d*x])^3*(102*(c + d*x) - 69*Cos[c + d*x] + Cos[3*(c + d*x) ] + 8/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (16*Sin[(c + d*x)/2])/(Cos [(c + d*x)/2] - Sin[(c + d*x)/2])^3 - (200*Sin[(c + d*x)/2])/(Cos[(c + d*x )/2] - Sin[(c + d*x)/2]) - 9*Sin[2*(c + d*x)]))/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.40 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3188, 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.
| \(\displaystyle \int \tan ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^4 (a \sin (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle a^4 \int \left (\frac {\sin ^3(c+d x)}{a}+\frac {3 \sin ^2(c+d x)}{a}+\frac {5 \sin (c+d x)}{a}+\frac {7}{a}-\frac {9}{a (1-\sin (c+d x))}+\frac {2}{a (1-\sin (c+d x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 \left (\frac {\cos ^3(c+d x)}{3 a d}-\frac {6 \cos (c+d x)}{a d}-\frac {3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {25 \cos (c+d x)}{3 a d (1-\sin (c+d x))}+\frac {2 \cos (c+d x)}{3 a d (1-\sin (c+d x))^2}+\frac {17 x}{2 a}\right )\) |
Input:
Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x]^4,x]
Output:
a^4*((17*x)/(2*a) - (6*Cos[c + d*x])/(a*d) + Cos[c + d*x]^3/(3*a*d) + (2*C os[c + d*x])/(3*a*d*(1 - Sin[c + d*x])^2) - (25*Cos[c + d*x])/(3*a*d*(1 - Sin[c + d*x])) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[a^p Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2)), x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
Result contains complex when optimal does not.
Time = 8.33 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(\frac {17 a^{3} x}{2}+\frac {3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {23 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {23 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 \left (-48 i a^{3} {\mathrm e}^{i \left (d x +c \right )}+27 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-25 a^{3}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {a^{3} \cos \left (3 d x +3 c \right )}{12 d}\) | \(149\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+a^{3} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(266\) |
| default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+a^{3} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(266\) |
| parts | \(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(277\) |
Input:
int((a+a*sin(d*x+c))^3*tan(d*x+c)^4,x,method=_RETURNVERBOSE)
Output:
17/2*a^3*x+3/8*I*a^3/d*exp(2*I*(d*x+c))-23/8*a^3/d*exp(I*(d*x+c))-23/8*a^3 /d*exp(-I*(d*x+c))-3/8*I*a^3/d*exp(-2*I*(d*x+c))-2/3*(-48*I*a^3*exp(I*(d*x +c))+27*a^3*exp(2*I*(d*x+c))-25*a^3)/(exp(I*(d*x+c))-I)^3/d+1/12*a^3/d*cos (3*d*x+3*c)
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (105) = 210\).
Time = 0.12 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.85 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {2 \, a^{3} \cos \left (d x + c\right )^{5} + 7 \, a^{3} \cos \left (d x + c\right )^{4} - 22 \, a^{3} \cos \left (d x + c\right )^{3} - 102 \, a^{3} d x - 4 \, a^{3} + {\left (51 \, a^{3} d x + 77 \, a^{3}\right )} \cos \left (d x + c\right )^{2} - {\left (51 \, a^{3} d x - 100 \, a^{3}\right )} \cos \left (d x + c\right ) + {\left (2 \, a^{3} \cos \left (d x + c\right )^{4} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + 102 \, a^{3} d x - 27 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (51 \, a^{3} d x - 104 \, a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \] Input:
integrate((a+a*sin(d*x+c))^3*tan(d*x+c)^4,x, algorithm="fricas")
Output:
1/6*(2*a^3*cos(d*x + c)^5 + 7*a^3*cos(d*x + c)^4 - 22*a^3*cos(d*x + c)^3 - 102*a^3*d*x - 4*a^3 + (51*a^3*d*x + 77*a^3)*cos(d*x + c)^2 - (51*a^3*d*x - 100*a^3)*cos(d*x + c) + (2*a^3*cos(d*x + c)^4 - 5*a^3*cos(d*x + c)^3 + 1 02*a^3*d*x - 27*a^3*cos(d*x + c)^2 - 4*a^3 + (51*a^3*d*x - 104*a^3)*cos(d* x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)
\[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=a^{3} \left (\int 3 \sin {\left (c + d x \right )} \tan ^{4}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \tan ^{4}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \tan ^{4}{\left (c + d x \right )}\, dx + \int \tan ^{4}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((a+a*sin(d*x+c))**3*tan(d*x+c)**4,x)
Output:
a**3*(Integral(3*sin(c + d*x)*tan(c + d*x)**4, x) + Integral(3*sin(c + d*x )**2*tan(c + d*x)**4, x) + Integral(sin(c + d*x)**3*tan(c + d*x)**4, x) + Integral(tan(c + d*x)**4, x))
Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.39 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {2 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a^{3} + 3 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{3} + 2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} - 6 \, a^{3} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{6 \, d} \] Input:
integrate((a+a*sin(d*x+c))^3*tan(d*x+c)^4,x, algorithm="maxima")
Output:
1/6*(2*(cos(d*x + c)^3 - (9*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 - 9*cos(d*x + c))*a^3 + 3*(2*tan(d*x + c)^3 + 15*d*x + 15*c - 3*tan(d*x + c)/(tan(d*x + c)^2 + 1) - 12*tan(d*x + c))*a^3 + 2*(tan(d*x + c)^3 + 3*d*x + 3*c - 3* tan(d*x + c))*a^3 - 6*a^3*((6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*cos(d *x + c)))/d
Timed out. \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(d*x+c))^3*tan(d*x+c)^4,x, algorithm="giac")
Output:
Timed out
Time = 20.18 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.12 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {17\,a^3\,x}{2}+\frac {\frac {17\,a^3\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {51\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (153\,c+153\,d\,x-378\right )}{6}\right )-\frac {a^3\,\left (51\,c+51\,d\,x-160\right )}{6}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {51\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (153\,c+153\,d\,x-102\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (51\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (306\,c+306\,d\,x-306\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (51\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (306\,c+306\,d\,x-654\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (85\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (510\,c+510\,d\,x-578\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (85\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (510\,c+510\,d\,x-1022\right )}{6}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (102\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (612\,c+612\,d\,x-918\right )}{6}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (102\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (612\,c+612\,d\,x-1002\right )}{6}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3} \] Input:
int(tan(c + d*x)^4*(a + a*sin(c + d*x))^3,x)
Output:
(17*a^3*x)/2 + ((17*a^3*(c + d*x))/2 - tan(c/2 + (d*x)/2)*((51*a^3*(c + d* x))/2 - (a^3*(153*c + 153*d*x - 378))/6) - (a^3*(51*c + 51*d*x - 160))/6 + tan(c/2 + (d*x)/2)^8*((51*a^3*(c + d*x))/2 - (a^3*(153*c + 153*d*x - 102) )/6) - tan(c/2 + (d*x)/2)^7*(51*a^3*(c + d*x) - (a^3*(306*c + 306*d*x - 30 6))/6) + tan(c/2 + (d*x)/2)^2*(51*a^3*(c + d*x) - (a^3*(306*c + 306*d*x - 654))/6) + tan(c/2 + (d*x)/2)^6*(85*a^3*(c + d*x) - (a^3*(510*c + 510*d*x - 578))/6) - tan(c/2 + (d*x)/2)^3*(85*a^3*(c + d*x) - (a^3*(510*c + 510*d* x - 1022))/6) - tan(c/2 + (d*x)/2)^5*(102*a^3*(c + d*x) - (a^3*(612*c + 61 2*d*x - 918))/6) + tan(c/2 + (d*x)/2)^4*(102*a^3*(c + d*x) - (a^3*(612*c + 612*d*x - 1002))/6))/(d*(tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^2 + tan( c/2 + (d*x)/2)^3 - 1)^3)
Time = 0.18 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.03 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {a^{3} \left (2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )^{3}-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )+45 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} c +51 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} d x +80 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-2 \cos \left (d x +c \right ) \tan \left (d x +c \right )^{3}+6 \cos \left (d x +c \right ) \tan \left (d x +c \right )-45 \cos \left (d x +c \right ) c -51 \cos \left (d x +c \right ) d x -80 \cos \left (d x +c \right )+2 \sin \left (d x +c \right )^{6}+9 \sin \left (d x +c \right )^{5}+30 \sin \left (d x +c \right )^{4}-60 \sin \left (d x +c \right )^{3}-120 \sin \left (d x +c \right )^{2}+45 \sin \left (d x +c \right )+80\right )}{6 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int((a+a*sin(d*x+c))^3*tan(d*x+c)^4,x)
Output:
(a**3*(2*cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)**3 - 6*cos(c + d*x)*sin (c + d*x)**2*tan(c + d*x) + 45*cos(c + d*x)*sin(c + d*x)**2*c + 51*cos(c + d*x)*sin(c + d*x)**2*d*x + 80*cos(c + d*x)*sin(c + d*x)**2 - 2*cos(c + d* x)*tan(c + d*x)**3 + 6*cos(c + d*x)*tan(c + d*x) - 45*cos(c + d*x)*c - 51* cos(c + d*x)*d*x - 80*cos(c + d*x) + 2*sin(c + d*x)**6 + 9*sin(c + d*x)**5 + 30*sin(c + d*x)**4 - 60*sin(c + d*x)**3 - 120*sin(c + d*x)**2 + 45*sin( c + d*x) + 80))/(6*cos(c + d*x)*d*(sin(c + d*x)**2 - 1))