∫ 1____________ (a+a sin(e+fx))3(c−c sin(e+fx))5 dx

3.288 \(\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx\)

Optimal. Leaf size=131 \[ \frac {2 \tan ^5(e+f x)}{15 a^3 c^5 f}+\frac {4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac {2 \tan (e+f x)}{3 a^3 c^5 f}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2} \]

[Out]

1/9*sec(f*x+e)^5/a^3/c^3/f/(c-c*sin(f*x+e))^2+1/9*sec(f*x+e)^5/a^3/f/(c^5-c^5*sin(f*x+e))+2/3*tan(f*x+e)/a^3/c

^5/f+4/9*tan(f*x+e)^3/a^3/c^5/f+2/15*tan(f*x+e)^5/a^3/c^5/f

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Rubi [A] time = 0.17, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2736, 2672, 3767} \[ \frac {2 \tan ^5(e+f x)}{15 a^3 c^5 f}+\frac {4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac {2 \tan (e+f x)}{3 a^3 c^5 f}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2} \]

Antiderivative was successfully veriﬁed.

[In]

Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5),x]

[Out]

Sec[e + f*x]^5/(9*a^3*c^3*f*(c - c*Sin[e + f*x])^2) + Sec[e + f*x]^5/(9*a^3*f*(c^5 - c^5*Sin[e + f*x])) + (2*T

an[e + f*x])/(3*a^3*c^5*f) + (4*Tan[e + f*x]^3)/(9*a^3*c^5*f) + (2*Tan[e + f*x]^5)/(15*a^3*c^5*f)

Rule 2672

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*

Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*

Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m

, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]

Rule 2736

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di

st[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&

EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,

n, m] || LtQ[m, n, 0]))

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]

, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx &=\frac {\int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{a^3 c^3}\\ &=\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {7 \int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{9 a^3 c^4}\\ &=\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \int \sec ^6(e+f x) \, dx}{3 a^3 c^5}\\ &=\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}-\frac {2 \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{3 a^3 c^5 f}\\ &=\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a^3 c^5 f}+\frac {4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac {2 \tan ^5(e+f x)}{15 a^3 c^5 f}\\ \end {align*}

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Mathematica [A] time = 1.54, size = 213, normalized size = 1.63 \[ \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (46080 \sin (e+f x)+3500 \sin (2 (e+f x))+19456 \sin (3 (e+f x))+2800 \sin (4 (e+f x))+1024 \sin (5 (e+f x))+700 \sin (6 (e+f x))-1024 \sin (7 (e+f x))-7875 \cos (e+f x)+20480 \cos (2 (e+f x))-3325 \cos (3 (e+f x))+16384 \cos (4 (e+f x))-175 \cos (5 (e+f x))+4096 \cos (6 (e+f x))+175 \cos (7 (e+f x)))}{184320 f (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^5} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5),x]

[Out]

((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-7875*Cos[e + f*x] + 20480*Cos[2

*(e + f*x)] - 3325*Cos[3*(e + f*x)] + 16384*Cos[4*(e + f*x)] - 175*Cos[5*(e + f*x)] + 4096*Cos[6*(e + f*x)] +

175*Cos[7*(e + f*x)] + 46080*Sin[e + f*x] + 3500*Sin[2*(e + f*x)] + 19456*Sin[3*(e + f*x)] + 2800*Sin[4*(e + f

*x)] + 1024*Sin[5*(e + f*x)] + 700*Sin[6*(e + f*x)] - 1024*Sin[7*(e + f*x)]))/(184320*f*(a + a*Sin[e + f*x])^3

*(c - c*Sin[e + f*x])^5)

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fricas [A] time = 0.46, size = 133, normalized size = 1.02 \[ -\frac {32 \, \cos \left (f x + e\right )^{6} - 16 \, \cos \left (f x + e\right )^{4} - 4 \, \cos \left (f x + e\right )^{2} - {\left (16 \, \cos \left (f x + e\right )^{6} - 24 \, \cos \left (f x + e\right )^{4} - 10 \, \cos \left (f x + e\right )^{2} - 7\right )} \sin \left (f x + e\right ) - 2}{45 \, {\left (a^{3} c^{5} f \cos \left (f x + e\right )^{7} + 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) - 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5}\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

-1/45*(32*cos(f*x + e)^6 - 16*cos(f*x + e)^4 - 4*cos(f*x + e)^2 - (16*cos(f*x + e)^6 - 24*cos(f*x + e)^4 - 10*

cos(f*x + e)^2 - 7)*sin(f*x + e) - 2)/(a^3*c^5*f*cos(f*x + e)^7 + 2*a^3*c^5*f*cos(f*x + e)^5*sin(f*x + e) - 2*

a^3*c^5*f*cos(f*x + e)^5)

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giac [A] time = 0.25, size = 217, normalized size = 1.66 \[ -\frac {\frac {3 \, {\left (435 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1470 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2060 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1330 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 353\right )}}{a^{3} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {4455 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 26460 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 78120 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 137340 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 157374 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 118356 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 57744 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 16596 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2339}{a^{3} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}}}{2880 \, f} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

-1/2880*(3*(435*tan(1/2*f*x + 1/2*e)^4 + 1470*tan(1/2*f*x + 1/2*e)^3 + 2060*tan(1/2*f*x + 1/2*e)^2 + 1330*tan(

1/2*f*x + 1/2*e) + 353)/(a^3*c^5*(tan(1/2*f*x + 1/2*e) + 1)^5) + (4455*tan(1/2*f*x + 1/2*e)^8 - 26460*tan(1/2*

f*x + 1/2*e)^7 + 78120*tan(1/2*f*x + 1/2*e)^6 - 137340*tan(1/2*f*x + 1/2*e)^5 + 157374*tan(1/2*f*x + 1/2*e)^4

- 118356*tan(1/2*f*x + 1/2*e)^3 + 57744*tan(1/2*f*x + 1/2*e)^2 - 16596*tan(1/2*f*x + 1/2*e) + 2339)/(a^3*c^5*(

tan(1/2*f*x + 1/2*e) - 1)^9))/f

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maple [A] time = 0.29, size = 223, normalized size = 1.70 \[ \frac {-\frac {4}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {49}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {49}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {35}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {49}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {51}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {99}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {13}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {9}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {29}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f \,a^{3} c^{5}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x)

[Out]

2/f/a^3/c^5*(-2/9/(tan(1/2*f*x+1/2*e)-1)^9-1/(tan(1/2*f*x+1/2*e)-1)^8-5/2/(tan(1/2*f*x+1/2*e)-1)^7-49/12/(tan(

1/2*f*x+1/2*e)-1)^6-49/10/(tan(1/2*f*x+1/2*e)-1)^5-35/8/(tan(1/2*f*x+1/2*e)-1)^4-49/16/(tan(1/2*f*x+1/2*e)-1)^

3-51/32/(tan(1/2*f*x+1/2*e)-1)^2-99/128/(tan(1/2*f*x+1/2*e)-1)-1/40/(tan(1/2*f*x+1/2*e)+1)^5+1/16/(tan(1/2*f*x

+1/2*e)+1)^4-13/96/(tan(1/2*f*x+1/2*e)+1)^3+9/64/(tan(1/2*f*x+1/2*e)+1)^2-29/128/(tan(1/2*f*x+1/2*e)+1))

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maxima [B] time = 0.44, size = 610, normalized size = 4.66 \[ \frac {2 \, {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {80 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {190 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {50 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {269 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {96 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {516 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {354 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {69 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {240 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {30 \, \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {90 \, \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} + \frac {45 \, \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} + 10\right )}}{45 \, {\left (a^{3} c^{5} - \frac {4 \, a^{3} c^{5} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a^{3} c^{5} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {16 \, a^{3} c^{5} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {19 \, a^{3} c^{5} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {20 \, a^{3} c^{5} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {45 \, a^{3} c^{5} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {45 \, a^{3} c^{5} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {20 \, a^{3} c^{5} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {19 \, a^{3} c^{5} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {16 \, a^{3} c^{5} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {a^{3} c^{5} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} + \frac {4 \, a^{3} c^{5} \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} - \frac {a^{3} c^{5} \sin \left (f x + e\right )^{14}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{14}}\right )} f} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm="maxima")

[Out]

2/45*(5*sin(f*x + e)/(cos(f*x + e) + 1) - 80*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 190*sin(f*x + e)^3/(cos(f*x

+ e) + 1)^3 + 50*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 269*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 96*sin(f*x +

e)^6/(cos(f*x + e) + 1)^6 + 516*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 354*sin(f*x + e)^8/(cos(f*x + e) + 1)^8

- 69*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 240*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 30*sin(f*x + e)^11/(co

s(f*x + e) + 1)^11 - 90*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 + 45*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 10)

/((a^3*c^5 - 4*a^3*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + a^3*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 16*a^3*

c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 19*a^3*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 20*a^3*c^5*sin(f*x

+ e)^5/(cos(f*x + e) + 1)^5 + 45*a^3*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 45*a^3*c^5*sin(f*x + e)^8/(cos(

f*x + e) + 1)^8 + 20*a^3*c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 19*a^3*c^5*sin(f*x + e)^10/(cos(f*x + e) +

1)^10 - 16*a^3*c^5*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - a^3*c^5*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 + 4*a

^3*c^5*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - a^3*c^5*sin(f*x + e)^14/(cos(f*x + e) + 1)^14)*f)

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mupad [B] time = 8.20, size = 190, normalized size = 1.45 \[ -\frac {\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {65\,\cos \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}{32}-\frac {225\,\cos \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}{32}-5\,\cos \left (\frac {7\,e}{2}+\frac {7\,f\,x}{2}\right )+\cos \left (\frac {9\,e}{2}+\frac {9\,f\,x}{2}\right )-\frac {37\,\cos \left (\frac {11\,e}{2}+\frac {11\,f\,x}{2}\right )}{32}+\frac {5\,\cos \left (\frac {13\,e}{2}+\frac {13\,f\,x}{2}\right )}{32}-\frac {89\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}+11\,\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )-\frac {63\,\sin \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}{8}+\frac {25\,\sin \left (\frac {7\,e}{2}+\frac {7\,f\,x}{2}\right )}{8}-\frac {5\,\sin \left (\frac {9\,e}{2}+\frac {9\,f\,x}{2}\right )}{8}+\frac {3\,\sin \left (\frac {11\,e}{2}+\frac {11\,f\,x}{2}\right )}{8}+\frac {\sin \left (\frac {13\,e}{2}+\frac {13\,f\,x}{2}\right )}{4}\right )}{2880\,a^3\,c^5\,f\,{\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}^5\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^5),x)

[Out]

-(cos(e/2 + (f*x)/2)*((65*cos((5*e)/2 + (5*f*x)/2))/32 - (225*cos((3*e)/2 + (3*f*x)/2))/32 - 5*cos((7*e)/2 + (

7*f*x)/2) + cos((9*e)/2 + (9*f*x)/2) - (37*cos((11*e)/2 + (11*f*x)/2))/32 + (5*cos((13*e)/2 + (13*f*x)/2))/32

- (89*sin(e/2 + (f*x)/2))/4 + 11*sin((3*e)/2 + (3*f*x)/2) - (63*sin((5*e)/2 + (5*f*x)/2))/8 + (25*sin((7*e)/2

+ (7*f*x)/2))/8 - (5*sin((9*e)/2 + (9*f*x)/2))/8 + (3*sin((11*e)/2 + (11*f*x)/2))/8 + sin((13*e)/2 + (13*f*x)/

2)/4))/(2880*a^3*c^5*f*cos(e/2 - pi/4 + (f*x)/2)^5*cos(e/2 + pi/4 + (f*x)/2)^9)

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sympy [A] time = 98.60, size = 4335, normalized size = 33.09 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**5,x)

[Out]

Piecewise((-90*tan(e/2 + f*x/2)**13/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**1

3 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x

/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/

2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*ta

n(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) +

180*tan(e/2 + f*x/2)**12/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3

*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 9

00*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)*

*6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x

/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 60*tan(e/2

+ f*x/2)**11/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(

e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5

*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**

3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*

a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 480*tan(e/2 + f*x/2)**1

0/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2

)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2

+ f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*ta

n(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f

*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) + 138*tan(e/2 + f*x/2)**9/(45*a**3*c

**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*

a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9

+ 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/

2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f

*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) + 708*tan(e/2 + f*x/2)**8/(45*a**3*c**5*f*tan(e/

2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*

tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*

c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*

a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 1

80*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 1032*tan(e/2 + f*x/2)**7/(45*a**3*c**5*f*tan(e/2 + f*x/2)*

*14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f

*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(

e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f

*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**

5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 192*tan(e/2 + f*x/2)**6/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a*

*3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 -

855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)

**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f

*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2

+ f*x/2) - 45*a**3*c**5*f) + 538*tan(e/2 + f*x/2)**5/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*ta

n(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**

5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a

**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 7

20*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 4

5*a**3*c**5*f) - 100*tan(e/2 + f*x/2)**4/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/

2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2

+ f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*t

an(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5

*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*

f) - 380*tan(e/2 + f*x/2)**3/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*

a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10

- 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x

/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 +

f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) + 160*tan

(e/2 + f*x/2)**2/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*

tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*

c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900

*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 -

45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 10*tan(e/2 + f*x/2)

/(45*a**3*c**5*f*tan(e/2 + f*x/2)**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)

**12 + 720*a**3*c**5*f*tan(e/2 + f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 +

f*x/2)**9 + 2025*a**3*c**5*f*tan(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan

(e/2 + f*x/2)**5 + 855*a**3*c**5*f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*

tan(e/2 + f*x/2)**2 + 180*a**3*c**5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f) - 20/(45*a**3*c**5*f*tan(e/2 + f*x/2)

**14 - 180*a**3*c**5*f*tan(e/2 + f*x/2)**13 + 45*a**3*c**5*f*tan(e/2 + f*x/2)**12 + 720*a**3*c**5*f*tan(e/2 +

f*x/2)**11 - 855*a**3*c**5*f*tan(e/2 + f*x/2)**10 - 900*a**3*c**5*f*tan(e/2 + f*x/2)**9 + 2025*a**3*c**5*f*tan

(e/2 + f*x/2)**8 - 2025*a**3*c**5*f*tan(e/2 + f*x/2)**6 + 900*a**3*c**5*f*tan(e/2 + f*x/2)**5 + 855*a**3*c**5*

f*tan(e/2 + f*x/2)**4 - 720*a**3*c**5*f*tan(e/2 + f*x/2)**3 - 45*a**3*c**5*f*tan(e/2 + f*x/2)**2 + 180*a**3*c*

*5*f*tan(e/2 + f*x/2) - 45*a**3*c**5*f), Ne(f, 0)), (x/((a*sin(e) + a)**3*(-c*sin(e) + c)**5), True))

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