4.5 Test file Number [79] 4-Trig-functions/4.1-Sine/4.1.7-d-trig-^m-a+b-c-sin-^n-^p

4.5.1 Mathematica

Integral number [399] $$\int \frac{{\cos }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 0.351766 (sec), size = 394 ,normalized size = 15.76 $$\frac{\frac{24\cos (c+dx)(a+b\sin (c+dx))}{4a+3b\sin (c+dx)-b\sin (3(c+dx))}-i\text{RootSum}[8{\mathrm{\#}\text{1}}^{3}a+i{\mathrm{\#}\text{1}}^{6}b-3i{\mathrm{\#}\text{1}}^{4}b+3i{\mathrm{\#}\text{1}}^{2}b-ib\mathrm{\&},\frac{-2{\mathrm{\#}\text{1}}^{3}a\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)+2\mathrm{\#}\text{1}a\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)-4i{\mathrm{\#}\text{1}}^{3}a{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)-i{\mathrm{\#}\text{1}}^{4}b\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)-6i{\mathrm{\#}\text{1}}^{2}b\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)-ib\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)+2{\mathrm{\#}\text{1}}^{4}b{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+12{\mathrm{\#}\text{1}}^{2}b{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+4i\mathrm{\#}\text{1}a{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+2b{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)}{-4i{\mathrm{\#}\text{1}}^{2}a+{\mathrm{\#}\text{1}}^{5}b-2{\mathrm{\#}\text{1}}^{3}b+\mathrm{\#}\text{1}b}\mathrm{\&}]}{18abd}$$

[In]

Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b*ArcTan[Sin[c + d*x]/(Cos[c +
d*x] - #1)] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2
*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 12*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (6*I)*b*Log[1 -
2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*Log[1 - 2*Cos[c
+ d*x]*#1 + #1^2]*#1^3 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #
1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (24*Cos[c + d*x]*(a + b*Sin[c + d*x]))/(4*a + 3*b*S
in[c + d*x] - b*Sin[3*(c + d*x)]))/(18*a*b*d)

Integral number [400] $$\int \frac{{\cos }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 0.245492 (sec), size = 273 ,normalized size = 10.92 $$\frac{\frac{12\sin (2(c+dx))}{4a+3b\sin (c+dx)-b\sin (3(c+dx))}-i\text{RootSum}[8{\mathrm{\#}\text{1}}^{3}a+i{\mathrm{\#}\text{1}}^{6}b-3i{\mathrm{\#}\text{1}}^{4}b+3i{\mathrm{\#}\text{1}}^{2}b-ib\mathrm{\&},\frac{-i{\mathrm{\#}\text{1}}^4\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)-6i{\mathrm{\#}\text{1}}^2\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)-i\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)+2{\mathrm{\#}\text{1}}^4{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+12{\mathrm{\#}\text{1}}^2{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+2{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)}{-4i{\mathrm{\#}\text{1}}^{2}a+{\mathrm{\#}\text{1}}^{5}b-2{\mathrm{\#}\text{1}}^{3}b+\mathrm{\#}\text{1}b}\mathrm{\&}]}{18ad}$$

[In]

Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d
*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 12*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (6*I)*Lo
g[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*Log[1 - 2*Cos[c + d
*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (12*Sin[2*(c + d*x)])/(4*a + 3*b*Sin[c +
d*x] - b*Sin[3*(c + d*x)]))/(18*a*d)

Integral number [401] $$\int \frac{1}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 0.459565 (sec), size = 502 ,normalized size = 31.38 $$\frac{-\frac{12b\cos (c+dx)(a\cos (2(c+dx))-3a+2b\sin (c+dx))}{(a-b)(a+b)(4a+3b\sin (c+dx)-b\sin (3(c+dx)))}+\frac{i\text{RootSum}[8{\mathrm{\#}\text{1}}^{3}a+i{\mathrm{\#}\text{1}}^{6}b-3i{\mathrm{\#}\text{1}}^{4}b+3i{\mathrm{\#}\text{1}}^{2}b-ib\mathrm{\&},\frac{12i{\mathrm{\#}\text{1}}^2{a}^2\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)-24{\mathrm{\#}\text{1}}^2{a}^2{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)-2{\mathrm{\#}\text{1}}^{3}ab\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)+2\mathrm{\#}\text{1}ab\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)-4i{\mathrm{\#}\text{1}}^{3}ab{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)-i{\mathrm{\#}\text{1}}^4{b}^2\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)-6i{\mathrm{\#}\text{1}}^2{b}^2\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)-i{b}^2\log ({\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\cos (c+dx)+1)+2{\mathrm{\#}\text{1}}^4{b}^2{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+12{\mathrm{\#}\text{1}}^2{b}^2{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+4i\mathrm{\#}\text{1}ab{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+2b^2{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)}{-4i{\mathrm{\#}\text{1}}^{2}a+{\mathrm{\#}\text{1}}^{5}b-2{\mathrm{\#}\text{1}}^{3}b+\mathrm{\#}\text{1}b}\mathrm{\&}]}{a^2-b^2}}{18ad}$$

[In]

Integrate[(a + b*Sin[c + d*x]^3)^(-2),x]

[Out]

((I*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b^2*ArcTan[Sin[c + d*x]/(Cos[c +
d*x] - #1)] - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1
+ 2*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 24*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^2*
ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (12*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^2*
Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*b*Log[1
- 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b^2*Log[1 - 2*Cos[
c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a^2 - b^2) - (12*b*Cos[c + d*x]*(-3*
a + a*Cos[2*(c + d*x)] + 2*b*Sin[c + d*x]))/((a - b)*(a + b)*(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)])))/(
18*a*d)

Integral number [402] $$\int \frac{{\sec }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 1.59132 (sec), size = 845 ,normalized size = 33.8 $$\frac{-\frac{ib\text{RootSum}[ib{\mathrm{\#}\text{1}}^6-3ib{\mathrm{\#}\text{1}}^4+8a{\mathrm{\#}\text{1}}^3+3ib{\mathrm{\#}\text{1}}^2-ib\mathrm{\&},\frac{2b^3{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^4+16a^{2}b{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^4-i{b}^3\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1){\mathrm{\#}\text{1}}^4-8i{a}^{2}b\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1){\mathrm{\#}\text{1}}^4-20i{a}^3{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^3-16ia{b}^2{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^3-10a^3\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1){\mathrm{\#}\text{1}}^3-8a{b}^2\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1){\mathrm{\#}\text{1}}^3+12b^3{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^2-120a^{2}b{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right){\mathrm{\#}\text{1}}^2-6i{b}^3\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1){\mathrm{\#}\text{1}}^2+60i{a}^{2}b\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1){\mathrm{\#}\text{1}}^2+20i{a}^3{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)\mathrm{\#}\text{1}+16ia{b}^2{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)\mathrm{\#}\text{1}+10a^3\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1)\mathrm{\#}\text{1}+8a{b}^2\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1)\mathrm{\#}\text{1}+2b^3{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)+16a^{2}b{\tan }^{-1}\left(\frac{\sin (c+dx)}{\cos (c+dx)-\mathrm{\#}\text{1}}\right)-i{b}^3\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1)-8i{a}^{2}b\log ({\mathrm{\#}\text{1}}^2-2\cos (c+dx)\mathrm{\#}\text{1}+1)}{b{\mathrm{\#}\text{1}}^5-2b{\mathrm{\#}\text{1}}^3-4ia{\mathrm{\#}\text{1}}^2+b\mathrm{\#}\text{1}}\mathrm{\&}]}{a{(a^2-b^2)}^2}+\frac{18\sin (\frac{1}{2}(c+dx))}{(a+b{)}^2(\cos (\frac{1}{2}(c+dx))-\sin (\frac{1}{2}(c+dx)))}+\frac{18\sin (\frac{1}{2}(c+dx))}{(a-b{)}^2(\cos (\frac{1}{2}(c+dx))+\sin (\frac{1}{2}(c+dx)))}+\frac{12b\cos (c+dx)(-2a^3-7b^{2}a+3b^2\cos (2(c+dx))a+2b(2a^2+b^2)\sin (c+dx))}{a(a-b{)}^2(a+b{)}^2(4a+3b\sin (c+dx)-b\sin (3(c+dx)))}}{18d}$$

[In]

Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

(((-I)*b*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (16*a^2*b*ArcTan[Sin[c + d*x]/
(Cos[c + d*x] - #1)] + 2*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1
+ #1^2] - I*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (
16*I)*a*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 8*a*b^
2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 120*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^3*ArcT
an[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (60*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^3*Lo
g[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (16*I)*a*b^2
*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 - 8*a*b^2*Log[1
- 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 16*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^3*ArcTan[Sin[
c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^3*Log[1 - 2*Cos[
c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a*(a^2 - b^2)^2) + (18*Sin[(c + d*x)
/2])/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (18*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])) + (12*b*Cos[c + d*x]*(-2*a^3 - 7*a*b^2 + 3*a*b^2*Cos[2*(c + d*x)] + 2*b*(2*a^2 + b^2)*Sin[c
+ d*x]))/(a*(a - b)^2*(a + b)^2*(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)])))/(18*d)

Integral number [403] $$\int \frac{{\sec }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[C]   time = 1.70294 (sec), size = 1158 ,normalized size = 46.32 $$\text{result too large to display}$$

[In]

Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((4*I)*b^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (14*a^4*ArcTan[Sin[c + d*x]/
(Cos[c + d*x] - #1)] + 74*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c
+ d*x] - #1)] - (7*I)*a^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]
- I*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (144*I)*a^3*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (36*I
)*a*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 18*a*b^3
*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 180*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 372*a^2*b^2*Ar
cTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (90*I)*a^
4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (186*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^4*
Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (144*I)*a^3*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (36*I)*
a*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 - 18*a*b
^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 14*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 74*a^2*b^2*
ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (7*I)*a^
4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^4*Log[1
- 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (3*Sec[c + d*x]^3*(48*a^5*b
+ 568*a^3*b^3 + 14*a*b^5 + (78*a^5*b + 606*a^3*b^3 + 81*a*b^5)*Cos[2*(c + d*x)] + 18*a*b^3*(4*a^2 + b^2)*Cos[
4*(c + d*x)] + 2*a^5*b*Cos[6*(c + d*x)] - 30*a^3*b^3*Cos[6*(c + d*x)] - 17*a*b^5*Cos[6*(c + d*x)] + 48*a^6*Sin
[c + d*x] - 244*a^4*b^2*Sin[c + d*x] + 20*a^2*b^4*Sin[c + d*x] - 4*b^6*Sin[c + d*x] + 16*a^6*Sin[3*(c + d*x)]
- 194*a^4*b^2*Sin[3*(c + d*x)] - 86*a^2*b^4*Sin[3*(c + d*x)] - 6*b^6*Sin[3*(c + d*x)] - 14*a^4*b^2*Sin[5*(c +
d*x)] - 74*a^2*b^4*Sin[5*(c + d*x)] - 2*b^6*Sin[5*(c + d*x)]))/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/
(72*a*(a^2 - b^2)^3*d)

4.5.2 Maple

Integral number [399] $$\int \frac{{\cos }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 0.244 (sec), size = 550 ,normalized size = 22. $$-\frac{2}{3da}{(\tan (\frac{dx}{2}+\frac{c}{2}))}^5{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{2}{3bd}{(\tan (\frac{dx}{2}+\frac{c}{2}))}^4{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{8}{3da}{(\tan (\frac{dx}{2}+\frac{c}{2}))}^3{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{4}{3bd}{(\tan (\frac{dx}{2}+\frac{c}{2}))}^2{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{2}{3da}\tan (\frac{dx}{2}+\frac{c}{2}){({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{2}{3bd}{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{2}{9abd}\sum _{\mathit{\_}\mathit{R}=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(a{\mathit{\_}\mathit{Z}}^6+3a{\mathit{\_}\mathit{Z}}^4+8b{\mathit{\_}\mathit{Z}}^3+3a{\mathit{\_}\mathit{Z}}^2+a)}\frac{{\mathit{\_}\mathit{R}}^{4}b+{\mathit{\_}\mathit{R}}^{3}a+\mathit{\_}\mathit{R}a+b}{{\mathit{\_}\mathit{R}}^{5}a+2{\mathit{\_}\mathit{R}}^{3}a+4{\mathit{\_}\mathit{R}}^{2}b+\mathit{\_}\mathit{R}a}\ln (\tan (\frac{dx}{2}+\frac{c}{2})-\mathit{\_}\mathit{R})$$

[In]

int(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x)

[Out]

-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a
*tan(1/2*d*x+1/2*c)^5+2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/
2*d*x+1/2*c)^2*a+a)/b*tan(1/2*d*x+1/2*c)^4+8/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*
d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)^3+4/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/
2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/b*tan(1/2*d*x+1/2*c)^2+2/3/d/(tan(1/2*d*x+1/2*c)
^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)+2/3/d/
(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/b+2/9/d/
a/b*sum((_R^4*b+_R^3*a+_R*a+b)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z
^4*a+8*_Z^3*b+3*_Z^2*a+a))

Integral number [400] $$\int \frac{{\cos }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 0.247 (sec), size = 236 ,normalized size = 9.44 $$-\frac{2}{3da}{(\tan (\frac{dx}{2}+\frac{c}{2}))}^5{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{2}{3da}\tan (\frac{dx}{2}+\frac{c}{2}){({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{2}{9da}\sum _{\mathit{\_}\mathit{R}=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(a{\mathit{\_}\mathit{Z}}^6+3a{\mathit{\_}\mathit{Z}}^4+8b{\mathit{\_}\mathit{Z}}^3+3a{\mathit{\_}\mathit{Z}}^2+a)}\frac{{\mathit{\_}\mathit{R}}^4+1}{{\mathit{\_}\mathit{R}}^{5}a+2{\mathit{\_}\mathit{R}}^{3}a+4{\mathit{\_}\mathit{R}}^{2}b+\mathit{\_}\mathit{R}a}\ln (\tan (\frac{dx}{2}+\frac{c}{2})-\mathit{\_}\mathit{R})$$

[In]

int(cos(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x)

[Out]

-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a
*tan(1/2*d*x+1/2*c)^5+2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/
2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)+2/9/d/a*sum((_R^4+1)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1
/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

Integral number [401] $$\int \frac{1}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 0.198 (sec), size = 658 ,normalized size = 41.12 $$\frac{2b^2}{3da(a^2-b^2)}{(\tan (\frac{dx}{2}+\frac{c}{2}))}^5{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}-\frac{2b}{3d(a^2-b^2)}{(\tan (\frac{dx}{2}+\frac{c}{2}))}^4{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{8b^2}{3da(a^2-b^2)}{(\tan (\frac{dx}{2}+\frac{c}{2}))}^3{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{8b}{3d(a^2-b^2)}{(\tan (\frac{dx}{2}+\frac{c}{2}))}^2{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}-\frac{2b^2}{3da(a^2-b^2)}\tan (\frac{dx}{2}+\frac{c}{2}){({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{2b}{3d(a^2-b^2)}{({(\tan (\frac{dx}{2}+\frac{c}{2}))}^{6}a+3{(\tan (1/2dx+c/2))}^{4}a+8b{(\tan (1/2dx+c/2))}^3+3{(\tan (1/2dx+c/2))}^{2}a+a)}^{-1}+\frac{1}{9da(a^2-b^2)}\sum _{\mathit{\_}\mathit{R}=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(a{\mathit{\_}\mathit{Z}}^6+3a{\mathit{\_}\mathit{Z}}^4+8b{\mathit{\_}\mathit{Z}}^3+3a{\mathit{\_}\mathit{Z}}^2+a)}\frac{(3a^2-2b^2){\mathit{\_}\mathit{R}}^4-2{\mathit{\_}\mathit{R}}^{3}ab+6{\mathit{\_}\mathit{R}}^2{a}^2-2\mathit{\_}\mathit{R}ab+3a^2-2b^2}{{\mathit{\_}\mathit{R}}^{5}a+2{\mathit{\_}\mathit{R}}^{3}a+4{\mathit{\_}\mathit{R}}^{2}b+\mathit{\_}\mathit{R}a}\ln (\tan (\frac{dx}{2}+\frac{c}{2})-\mathit{\_}\mathit{R})$$

[In]

int(1/(a+b*sin(d*x+c)^3)^2,x)

[Out]

2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*b^
2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)^5-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*
c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/(a^2-b^2)*b*tan(1/2*d*x+1/2*c)^4+8/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+
1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)^3+8/3/d/(ta
n(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/(a^2-b^2)*b
*tan(1/2*d*x+1/2*c)^2-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/
2*d*x+1/2*c)^2*a+a)*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)+2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+
8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/(a^2-b^2)*b+1/9/d/a/(a^2-b^2)*sum(((3*a^2-2*b^2)*_R^4-2*_
R^3*a*b+6*_R^2*a^2-2*_R*a*b+3*a^2-2*b^2)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_
Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

Integral number [402] $$\int \frac{{\sec }^2(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 0.269 (sec), size = 1276 ,normalized size = 51.04 $$\text{result too large to display}$$

[In]

int(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x)

[Out]

-4/3/d*b^2/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2
*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)^5-2/3/d*b^4/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*
c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)^5-2/3/d*b/(a-b)^2/(a+b)^2/(ta
n(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x
+1/2*c)^4*a^2+8/3/d*b^3/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c
)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^4-8/3/d*b^2/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1
/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)^3-16/3/d*b^4/(a-b)
^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+
a)/a*tan(1/2*d*x+1/2*c)^3-4/3/d*b/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2
*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^2*a^2-20/3/d*b^3/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2
*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^2+4/3
/d*b^2/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x
+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)+2/3/d*b^4/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a
+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)-2/3/d*b/(a-b)^2/(a+b)^2/(tan(1/2*d*
x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a^2-4/3/d*b^3/(a-b)
^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+
a)-1/9/d*b/(a-b)^2/(a+b)^2/a*sum((b*(11*a^2-2*b^2)*_R^4+2*a*(-5*a^2-4*b^2)*_R^3+54*_R^2*a^2*b+2*a*(-5*a^2-4*b^
2)*_R+11*a^2*b-2*b^3)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z
^3*b+3*_Z^2*a+a))-1/d/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)-1/d/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)

Integral number [403] $$\int \frac{{\sec }^4(c+dx)}{{(a+b{\sin }^3(c+dx))}^2}dx$$

[B]   time = 0.323 (sec), size = 1549 ,normalized size = 61.96 $$\text{result too large to display}$$

[In]

int(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x)

[Out]

2/3/d*b^2/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*
d*x+1/2*c)^2*a+a)*a^3*tan(1/2*d*x+1/2*c)^5+14/3/d*b^4/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/
2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)^5+2/3/d*b^6/(a-b)^3/(a+b)^3
/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1
/2*d*x+1/2*c)^5-6/d*b^5/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c
)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^4+16/d*b^4/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/
2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)^3+8/d*b^6/(a-b)^3/(
a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a
*tan(1/2*d*x+1/2*c)^3+12/d*b^3/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*
x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^2*a^2+12/d*b^5/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6
*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^2-2/3/d*b^
2/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*
c)^2*a+a)*a^3*tan(1/2*d*x+1/2*c)-14/3/d*b^4/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8
*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)-2/3/d*b^6/(a-b)^3/(a+b)^3/(tan(1/2*d*
x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c
)+4/d*b^3/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*
d*x+1/2*c)^2*a+a)*a^2+2/d*b^5/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x
+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)+1/9/d*b^2/(a-b)^3/(a+b)^3/a*sum(((19*a^4+28*a^2*b^2-2*b^4)*_R^4+18*a*b*(
-4*a^2-b^2)*_R^3+6*a^2*(11*a^2+34*b^2)*_R^2+18*a*b*(-4*a^2-b^2)*_R+19*a^4+28*a^2*b^2-2*b^4)/(_R^5*a+2*_R^3*a+4
*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))-1/3/d/(a+b)^2/(tan(1/2
*d*x+1/2*c)-1)^3-1/2/d/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)^2-1/d/(a+b)^3/(tan(1/2*d*x+1/2*c)-1)*a-4/d/(a+b)^3/(tan(
1/2*d*x+1/2*c)-1)*b-1/3/d/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)^3+1/2/d/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)^2+4/d/(a-b)^3/
(tan(1/2*d*x+1/2*c)+1)*b-1/d/(a-b)^3/(tan(1/2*d*x+1/2*c)+1)*a