3.3.78 \(\int \frac {1}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx\) [278]

Optimal. Leaf size=144 \[ \frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {8 \tan (e+f x)}{21 a^2 c^5 f}+\frac {8 \tan ^3(e+f x)}{63 a^2 c^5 f} \]

[Out]

1/9*sec(f*x+e)^3/a^2/c^2/f/(c-c*sin(f*x+e))^3+2/21*sec(f*x+e)^3/a^2/c^3/f/(c-c*sin(f*x+e))^2+2/21*sec(f*x+e)^3
/a^2/f/(c^5-c^5*sin(f*x+e))+8/21*tan(f*x+e)/a^2/c^5/f+8/63*tan(f*x+e)^3/a^2/c^5/f

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Rubi [A]
time = 0.15, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751, 3852} \begin {gather*} \frac {8 \tan ^3(e+f x)}{63 a^2 c^5 f}+\frac {8 \tan (e+f x)}{21 a^2 c^5 f}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5),x]

[Out]

Sec[e + f*x]^3/(9*a^2*c^2*f*(c - c*Sin[e + f*x])^3) + (2*Sec[e + f*x]^3)/(21*a^2*c^3*f*(c - c*Sin[e + f*x])^2)
+ (2*Sec[e + f*x]^3)/(21*a^2*f*(c^5 - c^5*Sin[e + f*x])) + (8*Tan[e + f*x])/(21*a^2*c^5*f) + (8*Tan[e + f*x]^
3)/(63*a^2*c^5*f)

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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 2815

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 3852

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))^2 (c-c \sin (e+f x))^5} \, dx &=\frac {\int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{a^2 c^2}\\ &=\frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{3 a^2 c^3}\\ &=\frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {10 \int \frac {\sec ^4(e+f x)}{c-c \sin (e+f x)} \, dx}{21 a^2 c^4}\\ &=\frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {8 \int \sec ^4(e+f x) \, dx}{21 a^2 c^5}\\ &=\frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}-\frac {8 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{21 a^2 c^5 f}\\ &=\frac {\sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {2 \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {2 \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {8 \tan (e+f x)}{21 a^2 c^5 f}+\frac {8 \tan ^3(e+f x)}{63 a^2 c^5 f}\\ \end {align*}

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Mathematica [A]
time = 0.78, size = 193, normalized size = 1.34 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-5580 \cos (e+f x)+13824 \cos (2 (e+f x))-310 \cos (3 (e+f x))+6144 \cos (4 (e+f x))+930 \cos (5 (e+f x))-512 \cos (6 (e+f x))+18432 \sin (e+f x)+4185 \sin (2 (e+f x))+1024 \sin (3 (e+f x))+1860 \sin (4 (e+f x))-3072 \sin (5 (e+f x))-155 \sin (6 (e+f x)))}{64512 f (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sin[e + f*x])^2*(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])*(-5580*Cos[e + f*x] + 13824*Cos[2
*(e + f*x)] - 310*Cos[3*(e + f*x)] + 6144*Cos[4*(e + f*x)] + 930*Cos[5*(e + f*x)] - 512*Cos[6*(e + f*x)] + 184
32*Sin[e + f*x] + 4185*Sin[2*(e + f*x)] + 1024*Sin[3*(e + f*x)] + 1860*Sin[4*(e + f*x)] - 3072*Sin[5*(e + f*x)
] - 155*Sin[6*(e + f*x)]))/(64512*f*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5)

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Maple [A]
time = 0.44, size = 193, normalized size = 1.34

method	result	size
risch	\(-\frac {32 \left (-6 \,{\mathrm e}^{i \left (f x +e \right )}+2 \,{\mathrm e}^{3 i \left (f x +e \right )}+36 \,{\mathrm e}^{5 i \left (f x +e \right )}+i-27 i {\mathrm e}^{4 i \left (f x +e \right )}-12 i {\mathrm e}^{2 i \left (f x +e \right )}\right )}{63 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f \,c^{5} a^{2}}\)	\(100\)
derivativedivides	\(\frac {-\frac {8}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {59}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {9}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{24 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{2} f \,c^{5}}\)	\(193\)
default	\(\frac {-\frac {8}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {68}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {35}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {59}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {19}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {9}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {57}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{24 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {7}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{a^{2} f \,c^{5}}\)	\(193\)
norman	\(\frac {-\frac {28 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {38}{63 a c f}+\frac {6 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {12 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {2 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {26 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {470 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 a c f}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{21 a c f}-\frac {68 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}-\frac {26 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 a c f}+\frac {100 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 a c f}}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

2/f/a^2/c^5*(-4/9/(tan(1/2*f*x+1/2*e)-1)^9-2/(tan(1/2*f*x+1/2*e)-1)^8-34/7/(tan(1/2*f*x+1/2*e)-1)^7-23/3/(tan(
1/2*f*x+1/2*e)-1)^6-35/4/(tan(1/2*f*x+1/2*e)-1)^5-59/8/(tan(1/2*f*x+1/2*e)-1)^4-19/4/(tan(1/2*f*x+1/2*e)-1)^3-
9/4/(tan(1/2*f*x+1/2*e)-1)^2-57/64/(tan(1/2*f*x+1/2*e)-1)-1/48/(tan(1/2*f*x+1/2*e)+1)^3+1/32/(tan(1/2*f*x+1/2*
e)+1)^2-7/64/(tan(1/2*f*x+1/2*e)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (145) = 290\).
time = 0.34, size = 563, normalized size = 3.91 \begin {gather*} -\frac {2 \, {\left (\frac {51 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {39 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {235 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {450 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {306 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {294 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {378 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {273 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {189 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {63 \, \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - 19\right )}}{63 \, {\left (a^{2} c^{5} - \frac {6 \, a^{2} c^{5} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {12 \, a^{2} c^{5} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2 \, a^{2} c^{5} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {27 \, a^{2} c^{5} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {36 \, a^{2} c^{5} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {36 \, a^{2} c^{5} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {27 \, a^{2} c^{5} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {2 \, a^{2} c^{5} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - \frac {12 \, a^{2} c^{5} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {6 \, a^{2} c^{5} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {a^{2} c^{5} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}}\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="maxima")

[Out]

-2/63*(51*sin(f*x + e)/(cos(f*x + e) + 1) - 39*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 235*sin(f*x + e)^3/(cos(f
*x + e) + 1)^3 + 450*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 306*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 294*sin(f
*x + e)^6/(cos(f*x + e) + 1)^6 + 378*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 63*sin(f*x + e)^8/(cos(f*x + e) + 1
)^8 - 273*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 189*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 63*sin(f*x + e)^11
/(cos(f*x + e) + 1)^11 - 19)/((a^2*c^5 - 6*a^2*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^2*c^5*sin(f*x + e)^2
/(cos(f*x + e) + 1)^2 - 2*a^2*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 27*a^2*c^5*sin(f*x + e)^4/(cos(f*x + e
) + 1)^4 + 36*a^2*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 36*a^2*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 2
7*a^2*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2*a^2*c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 12*a^2*c^5*sin
(f*x + e)^10/(cos(f*x + e) + 1)^10 + 6*a^2*c^5*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - a^2*c^5*sin(f*x + e)^12
/(cos(f*x + e) + 1)^12)*f)

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Fricas [A]
time = 0.32, size = 154, normalized size = 1.07 \begin {gather*} \frac {16 \, \cos \left (f x + e\right )^{6} - 72 \, \cos \left (f x + e\right )^{4} + 30 \, \cos \left (f x + e\right )^{2} + 2 \, {\left (24 \, \cos \left (f x + e\right )^{4} - 20 \, \cos \left (f x + e\right )^{2} - 7\right )} \sin \left (f x + e\right ) + 7}{63 \, {\left (3 \, a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3} - {\left (a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

1/63*(16*cos(f*x + e)^6 - 72*cos(f*x + e)^4 + 30*cos(f*x + e)^2 + 2*(24*cos(f*x + e)^4 - 20*cos(f*x + e)^2 - 7
)*sin(f*x + e) + 7)/(3*a^2*c^5*f*cos(f*x + e)^5 - 4*a^2*c^5*f*cos(f*x + e)^3 - (a^2*c^5*f*cos(f*x + e)^5 - 4*a
^2*c^5*f*cos(f*x + e)^3)*sin(f*x + e))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3186 vs. \(2 (131) = 262\).
time = 28.05, size = 3186, normalized size = 22.12 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))**2/(c-c*sin(f*x+e))**5,x)

[Out]

Piecewise((-126*tan(e/2 + f*x/2)**11/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**
11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f
*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(
e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*
tan(e/2 + f*x/2) - 63*a**2*c**5*f) + 378*tan(e/2 + f*x/2)**10/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*
c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 170
1*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**
5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x
/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 546*tan(e/2 + f*x/2)**9/(63*a**2*c**5*f*tan(e/2
+ f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*t
an(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c
**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a
**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 126*tan(e/2 + f*x/2)**8/
(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)
**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 +
f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan
(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) +
756*tan(e/2 + f*x/2)**7/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2
*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 22
68*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)*
*4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x
/2) - 63*a**2*c**5*f) - 588*tan(e/2 + f*x/2)**6/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2
+ f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*
tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*
c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a
**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 612*tan(e/2 + f*x/2)**5/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 -
378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2
)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2
+ f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan
(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) + 900*tan(e/2 + f*x/2)**4/(63*a**2*c**5*
f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**
2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2
268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)*
*3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 470*tan(e/2 +
f*x/2)**3/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/
2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f
*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*
c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*
c**5*f) - 78*tan(e/2 + f*x/2)**2/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 +
756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2
)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2
+ f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(
e/2 + f*x/2) - 63*a**2*c**5*f) + 102*tan(e/2 + f*x/2)/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*t
an(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c
**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701
*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**...

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Giac [A]
time = 0.48, size = 189, normalized size = 1.31 \begin {gather*} -\frac {\frac {21 \, {\left (21 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 19\right )}}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {3591 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 19656 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 56196 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 95760 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 107730 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 79464 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 38484 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10944 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1615}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}}}{2016 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

-1/2016*(21*(21*tan(1/2*f*x + 1/2*e)^2 + 36*tan(1/2*f*x + 1/2*e) + 19)/(a^2*c^5*(tan(1/2*f*x + 1/2*e) + 1)^3)
+ (3591*tan(1/2*f*x + 1/2*e)^8 - 19656*tan(1/2*f*x + 1/2*e)^7 + 56196*tan(1/2*f*x + 1/2*e)^6 - 95760*tan(1/2*f
*x + 1/2*e)^5 + 107730*tan(1/2*f*x + 1/2*e)^4 - 79464*tan(1/2*f*x + 1/2*e)^3 + 38484*tan(1/2*f*x + 1/2*e)^2 -
10944*tan(1/2*f*x + 1/2*e) + 1615)/(a^2*c^5*(tan(1/2*f*x + 1/2*e) - 1)^9))/f

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Mupad [B]
time = 9.29, size = 180, normalized size = 1.25 \begin {gather*} -\frac {2\,\left (63\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}-189\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+273\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+63\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-378\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+294\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+306\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-450\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+235\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+39\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-51\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+19\right )}{63\,a^2\,c^5\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^9\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^5),x)

[Out]

-(2*(39*tan(e/2 + (f*x)/2)^2 - 51*tan(e/2 + (f*x)/2) + 235*tan(e/2 + (f*x)/2)^3 - 450*tan(e/2 + (f*x)/2)^4 + 3
06*tan(e/2 + (f*x)/2)^5 + 294*tan(e/2 + (f*x)/2)^6 - 378*tan(e/2 + (f*x)/2)^7 + 63*tan(e/2 + (f*x)/2)^8 + 273*
tan(e/2 + (f*x)/2)^9 - 189*tan(e/2 + (f*x)/2)^10 + 63*tan(e/2 + (f*x)/2)^11 + 19))/(63*a^2*c^5*f*(tan(e/2 + (f
*x)/2) - 1)^9*(tan(e/2 + (f*x)/2) + 1)^3)

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