3.6.87 \(\int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx\) [587]

Optimal. Leaf size=317 \[ \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]

[Out]

2/63*a^3*(c-d)*(3*c+19*d)*cos(f*x+e)/d^2/(c+d)^2/f/(c+d*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(1/2)-2/105*a^3*(c^
2+10*c*d+73*d^2)*cos(f*x+e)/d^2/(c+d)^3/f/(c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)-8/315*a^3*(c^2+10*c*d+
73*d^2)*cos(f*x+e)/d^2/(c+d)^4/f/(c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)+2/9*a^2*(c-d)*cos(f*x+e)*(a+a*s
in(f*x+e))^(1/2)/d/(c+d)/f/(c+d*sin(f*x+e))^(9/2)-16/315*a^3*(c^2+10*c*d+73*d^2)*cos(f*x+e)/d^2/(c+d)^5/f/(a+a
*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.51, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2841, 3059, 2851, 2850} \begin {gather*} -\frac {16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 f (c+d)^5 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 f (c+d)^4 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}+\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(11/2),x]

[Out]

(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(9*d*(c + d)*f*(c + d*Sin[e + f*x])^(9/2)) + (2*a^3*(c -
d)*(3*c + 19*d)*Cos[e + f*x])/(63*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a
^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5
/2)) - (8*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e
+ f*x])^(3/2)) - (16*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^5*f*Sqrt[a + a*Sin[e + f*x]]*
Sqrt[c + d*Sin[e + f*x]])

Rule 2841

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c
+ a*d))), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1
)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{
a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1
] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2850

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> Sim
p[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])), x] /; FreeQ[{a, b,
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2851

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x]
+ Dist[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]

Rule 3059

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n +
1)*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}-\frac {(2 a) \int \frac {\sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a (c-19 d)-\frac {3}{2} a (c+5 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{9/2}} \, dx}{9 d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}+\frac {\left (a^2 \left (c^2+10 c d+73 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx}{21 d^2 (c+d)^2}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac {\left (4 a^2 \left (c^2+10 c d+73 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{105 d^2 (c+d)^3}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {\left (8 a^2 \left (c^2+10 c d+73 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{315 d^2 (c+d)^4}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 5.21, size = 304, normalized size = 0.96 \begin {gather*} -\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (1869 c^4+2088 c^3 d+5776 c^2 d^2+1804 c d^3+727 d^4-\left (63 c^4+648 c^3 d+4790 c^2 d^2+1424 c d^3+803 d^4\right ) \cos (2 (e+f x))+2 d^2 \left (c^2+10 c d+73 d^2\right ) \cos (4 (e+f x))+588 c^4 \sin (e+f x)+7326 c^3 d \sin (e+f x)+4370 c^2 d^2 \sin (e+f x)+5498 c d^3 \sin (e+f x)+698 d^4 \sin (e+f x)-18 c^3 d \sin (3 (e+f x))-182 c^2 d^2 \sin (3 (e+f x))-1334 c d^3 \sin (3 (e+f x))-146 d^4 \sin (3 (e+f x))\right )}{315 (c+d)^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(11/2),x]

[Out]

-1/315*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(1869*c^4 + 2088*c^3*d + 5776*c^2
*d^2 + 1804*c*d^3 + 727*d^4 - (63*c^4 + 648*c^3*d + 4790*c^2*d^2 + 1424*c*d^3 + 803*d^4)*Cos[2*(e + f*x)] + 2*
d^2*(c^2 + 10*c*d + 73*d^2)*Cos[4*(e + f*x)] + 588*c^4*Sin[e + f*x] + 7326*c^3*d*Sin[e + f*x] + 4370*c^2*d^2*S
in[e + f*x] + 5498*c*d^3*Sin[e + f*x] + 698*d^4*Sin[e + f*x] - 18*c^3*d*Sin[3*(e + f*x)] - 182*c^2*d^2*Sin[3*(
e + f*x)] - 1334*c*d^3*Sin[3*(e + f*x)] - 146*d^4*Sin[3*(e + f*x)]))/((c + d)^5*f*(Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])*(c + d*Sin[e + f*x])^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1729\) vs. \(2(287)=574\).
time = 7.51, size = 1730, normalized size = 5.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1730\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x,method=_RETURNVERBOSE)

[Out]

-2/315/f*(a*(1+sin(f*x+e)))^(5/2)*(c+d*sin(f*x+e))^(1/2)*(4224*d*c^8-12288*d^3*c^6+13568*d^5*c^4+8*cos(f*x+e)^
12*c^2*d^7+584*cos(f*x+e)^12*d^9-4599*cos(f*x+e)^10*d^9+14245*cos(f*x+e)^8*d^9-22645*cos(f*x+e)^6*d^9-105*cos(
f*x+e)^4*c^9+19695*cos(f*x+e)^4*d^9-1680*cos(f*x+e)^2*c^9-8944*cos(f*x+e)^2*d^9-2688*sin(f*x+e)*c^9-1664*sin(f
*x+e)*d^9+1664*d^9-35*cos(f*x+e)^8*sin(f*x+e)*c^5*d^4+5*cos(f*x+e)^8*sin(f*x+e)*c^4*d^5+1650*cos(f*x+e)^8*sin(
f*x+e)*c^3*d^6+5850*cos(f*x+e)^8*sin(f*x+e)*c^2*d^7-3295*cos(f*x+e)^8*sin(f*x+e)*c*d^8+458*cos(f*x+e)^6*sin(f*
x+e)*c^7*d^2+2730*cos(f*x+e)^6*sin(f*x+e)*c^6*d^3+8774*cos(f*x+e)^6*sin(f*x+e)*c^5*d^4+17070*cos(f*x+e)^6*sin(
f*x+e)*c^4*d^5-6490*cos(f*x+e)^6*sin(f*x+e)*c^3*d^6-24522*cos(f*x+e)^6*sin(f*x+e)*c^2*d^7+8010*cos(f*x+e)^6*si
n(f*x+e)*c*d^8-15*cos(f*x+e)^4*sin(f*x+e)*c^8*d+1812*cos(f*x+e)^4*sin(f*x+e)*c^7*d^2+5380*cos(f*x+e)^4*sin(f*x
+e)*c^6*d^3-22482*cos(f*x+e)^4*sin(f*x+e)*c^5*d^4-44418*cos(f*x+e)^4*sin(f*x+e)*c^4*d^5+13860*cos(f*x+e)^4*sin
(f*x+e)*c^3*d^6+38772*cos(f*x+e)^4*sin(f*x+e)*c^2*d^7+2688*c^9+80*cos(f*x+e)^12*c*d^8+1460*cos(f*x+e)^10*sin(f
*x+e)*d^9+37*cos(f*x+e)^10*c^4*d^5+360*cos(f*x+e)^10*c^3*d^6+2538*cos(f*x+e)^10*c^2*d^7-1360*cos(f*x+e)^10*c*d
^8-7535*cos(f*x+e)^8*sin(f*x+e)*d^9+310*cos(f*x+e)^8*c^6*d^3+1875*cos(f*x+e)^8*c^5*d^4+6805*cos(f*x+e)^8*c^4*d
^5-2930*cos(f*x+e)^8*c^3*d^6-16320*cos(f*x+e)^8*c^2*d^7+6095*cos(f*x+e)^8*c*d^8+15474*cos(f*x+e)^6*sin(f*x+e)*
d^9-279*cos(f*x+e)^6*c^8*d-1310*cos(f*x+e)^6*c^7*d^2-1482*cos(f*x+e)^6*c^6*d^3-17010*cos(f*x+e)^6*c^5*d^4-3598
0*cos(f*x+e)^6*c^4*d^5+11406*cos(f*x+e)^6*c^3*d^6+41570*cos(f*x+e)^6*c^2*d^7-11902*cos(f*x+e)^6*c*d^8-63*cos(f
*x+e)^4*sin(f*x+e)*c^9-15847*cos(f*x+e)^4*sin(f*x+e)*d^9+87*cos(f*x+e)^4*c^8*d-7220*cos(f*x+e)^4*c^7*d^2-15204
*cos(f*x+e)^4*c^6*d^3+31650*cos(f*x+e)^4*c^5*d^4+63090*cos(f*x+e)^4*c^4*d^5-19908*cos(f*x+e)^4*c^3*d^6-51540*c
os(f*x+e)^4*c^2*d^7+11711*cos(f*x+e)^4*c*d^8+336*cos(f*x+e)^2*sin(f*x+e)*c^9+8112*cos(f*x+e)^2*sin(f*x+e)*d^9-
3312*cos(f*x+e)^2*c^8*d+16192*cos(f*x+e)^2*c^7*d^2+28864*cos(f*x+e)^2*c^6*d^3-23904*cos(f*x+e)^2*c^5*d^4-47520
*cos(f*x+e)^2*c^4*d^5+15168*cos(f*x+e)^2*c^3*d^6+30912*cos(f*x+e)^2*c^2*d^7-5776*cos(f*x+e)^2*c*d^8-4224*sin(f
*x+e)*c^8*d+7168*sin(f*x+e)*c^7*d^2+12288*sin(f*x+e)*c^6*d^3-7424*sin(f*x+e)*c^5*d^4-13568*sin(f*x+e)*c^4*d^5+
4096*sin(f*x+e)*c^3*d^6+7168*sin(f*x+e)*c^2*d^7-1152*sin(f*x+e)*c*d^8-7168*c^7*d^2+7424*c^5*d^4-4096*c^3*d^6+1
152*c*d^8+1200*cos(f*x+e)^2*sin(f*x+e)*c^8*d-12608*cos(f*x+e)^2*sin(f*x+e)*c^7*d^2-22720*cos(f*x+e)^2*sin(f*x+
e)*c^6*d^3+20192*cos(f*x+e)^2*sin(f*x+e)*c^5*d^4+40736*cos(f*x+e)^2*sin(f*x+e)*c^4*d^5-13120*cos(f*x+e)^2*sin(
f*x+e)*c^3*d^6-27328*cos(f*x+e)^2*sin(f*x+e)*c^2*d^7+5200*cos(f*x+e)^2*sin(f*x+e)*c*d^8-9255*cos(f*x+e)^4*sin(
f*x+e)*c*d^8+4*cos(f*x+e)^10*sin(f*x+e)*c^3*d^6+60*cos(f*x+e)^10*sin(f*x+e)*c^2*d^7+492*cos(f*x+e)^10*sin(f*x+
e)*c*d^8-7168*c^2*d^7)/cos(f*x+e)^5/(cos(f*x+e)^2*d^2+c^2-d^2)^5/(c+d)^5

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1018 vs. \(2 (302) = 604\).
time = 0.84, size = 1018, normalized size = 3.21 \begin {gather*} -\frac {2 \, {\left ({\left (903 \, c^{5} + 720 \, c^{4} d + 494 \, c^{3} d^{2} + 200 \, c^{2} d^{3} + 35 \, c d^{4}\right )} a^{\frac {5}{2}} - \frac {{\left (315 \, c^{5} - 8358 \, c^{4} d - 4770 \, c^{3} d^{2} - 2284 \, c^{2} d^{3} - 625 \, c d^{4} - 70 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (4179 \, c^{5} - 1710 \, c^{4} d + 30878 \, c^{3} d^{2} + 11540 \, c^{2} d^{3} + 3383 \, c d^{4} + 450 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {3 \, {\left (805 \, c^{5} - 9912 \, c^{4} d + 2330 \, c^{3} d^{2} - 18504 \, c^{2} d^{3} - 3895 \, c d^{4} - 504 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {6 \, {\left (1239 \, c^{5} - 3100 \, c^{4} d + 12918 \, c^{3} d^{2} - 3560 \, c^{2} d^{3} + 8043 \, c d^{4} + 700 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {42 \, {\left (149 \, c^{5} - 894 \, c^{4} d + 1402 \, c^{3} d^{2} - 2052 \, c^{2} d^{3} + 745 \, c d^{4} - 390 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {42 \, {\left (149 \, c^{5} - 894 \, c^{4} d + 1402 \, c^{3} d^{2} - 2052 \, c^{2} d^{3} + 745 \, c d^{4} - 390 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {6 \, {\left (1239 \, c^{5} - 3100 \, c^{4} d + 12918 \, c^{3} d^{2} - 3560 \, c^{2} d^{3} + 8043 \, c d^{4} + 700 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {3 \, {\left (805 \, c^{5} - 9912 \, c^{4} d + 2330 \, c^{3} d^{2} - 18504 \, c^{2} d^{3} - 3895 \, c d^{4} - 504 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {{\left (4179 \, c^{5} - 1710 \, c^{4} d + 30878 \, c^{3} d^{2} + 11540 \, c^{2} d^{3} + 3383 \, c d^{4} + 450 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {{\left (315 \, c^{5} - 8358 \, c^{4} d - 4770 \, c^{3} d^{2} - 2284 \, c^{2} d^{3} - 625 \, c d^{4} - 70 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {{\left (903 \, c^{5} + 720 \, c^{4} d + 494 \, c^{3} d^{2} + 200 \, c^{2} d^{3} + 35 \, c d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{3}}{315 \, {\left (c^{5} + 5 \, c^{4} d + 10 \, c^{3} d^{2} + 10 \, c^{2} d^{3} + 5 \, c d^{4} + d^{5} + \frac {3 \, {\left (c^{5} + 5 \, c^{4} d + 10 \, c^{3} d^{2} + 10 \, c^{2} d^{3} + 5 \, c d^{4} + d^{5}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, {\left (c^{5} + 5 \, c^{4} d + 10 \, c^{3} d^{2} + 10 \, c^{2} d^{3} + 5 \, c d^{4} + d^{5}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {{\left (c^{5} + 5 \, c^{4} d + 10 \, c^{3} d^{2} + 10 \, c^{2} d^{3} + 5 \, c d^{4} + d^{5}\right )} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )} {\left (c + \frac {2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac {11}{2}} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm="maxima")

[Out]

-2/315*((903*c^5 + 720*c^4*d + 494*c^3*d^2 + 200*c^2*d^3 + 35*c*d^4)*a^(5/2) - (315*c^5 - 8358*c^4*d - 4770*c^
3*d^2 - 2284*c^2*d^3 - 625*c*d^4 - 70*d^5)*a^(5/2)*sin(f*x + e)/(cos(f*x + e) + 1) + (4179*c^5 - 1710*c^4*d +
30878*c^3*d^2 + 11540*c^2*d^3 + 3383*c*d^4 + 450*d^5)*a^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 3*(805*c^5
- 9912*c^4*d + 2330*c^3*d^2 - 18504*c^2*d^3 - 3895*c*d^4 - 504*d^5)*a^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 + 6*(1239*c^5 - 3100*c^4*d + 12918*c^3*d^2 - 3560*c^2*d^3 + 8043*c*d^4 + 700*d^5)*a^(5/2)*sin(f*x + e)^4/(c
os(f*x + e) + 1)^4 - 42*(149*c^5 - 894*c^4*d + 1402*c^3*d^2 - 2052*c^2*d^3 + 745*c*d^4 - 390*d^5)*a^(5/2)*sin(
f*x + e)^5/(cos(f*x + e) + 1)^5 + 42*(149*c^5 - 894*c^4*d + 1402*c^3*d^2 - 2052*c^2*d^3 + 745*c*d^4 - 390*d^5)
*a^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6*(1239*c^5 - 3100*c^4*d + 12918*c^3*d^2 - 3560*c^2*d^3 + 8043*
c*d^4 + 700*d^5)*a^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3*(805*c^5 - 9912*c^4*d + 2330*c^3*d^2 - 18504*
c^2*d^3 - 3895*c*d^4 - 504*d^5)*a^(5/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - (4179*c^5 - 1710*c^4*d + 30878*c
^3*d^2 + 11540*c^2*d^3 + 3383*c*d^4 + 450*d^5)*a^(5/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + (315*c^5 - 8358*c
^4*d - 4770*c^3*d^2 - 2284*c^2*d^3 - 625*c*d^4 - 70*d^5)*a^(5/2)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - (903*
c^5 + 720*c^4*d + 494*c^3*d^2 + 200*c^2*d^3 + 35*c*d^4)*a^(5/2)*sin(f*x + e)^11/(cos(f*x + e) + 1)^11)*(sin(f*
x + e)^2/(cos(f*x + e) + 1)^2 + 1)^3/((c^5 + 5*c^4*d + 10*c^3*d^2 + 10*c^2*d^3 + 5*c*d^4 + d^5 + 3*(c^5 + 5*c^
4*d + 10*c^3*d^2 + 10*c^2*d^3 + 5*c*d^4 + d^5)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*(c^5 + 5*c^4*d + 10*c^3
*d^2 + 10*c^2*d^3 + 5*c*d^4 + d^5)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (c^5 + 5*c^4*d + 10*c^3*d^2 + 10*c^2*
d^3 + 5*c*d^4 + d^5)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6)*(c + 2*d*sin(f*x + e)/(cos(f*x + e) + 1) + c*sin(f*x
+ e)^2/(cos(f*x + e) + 1)^2)^(11/2)*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1516 vs. \(2 (302) = 604\).
time = 0.43, size = 1516, normalized size = 4.78 \begin {gather*} \frac {2 \, {\left (672 \, a^{2} c^{4} - 2304 \, a^{2} c^{3} d + 3008 \, a^{2} c^{2} d^{2} - 1792 \, a^{2} c d^{3} + 416 \, a^{2} d^{4} + 8 \, {\left (a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 73 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{5} - 4 \, {\left (9 \, a^{2} c^{3} d + 89 \, a^{2} c^{2} d^{2} + 647 \, a^{2} c d^{3} - 73 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{4} - {\left (63 \, a^{2} c^{4} + 648 \, a^{2} c^{3} d + 4798 \, a^{2} c^{2} d^{2} + 1504 \, a^{2} c d^{3} + 1387 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (231 \, a^{2} c^{4} + 3060 \, a^{2} c^{3} d - 2158 \, a^{2} c^{2} d^{2} + 4580 \, a^{2} c d^{3} - 673 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (483 \, a^{2} c^{4} + 684 \, a^{2} c^{3} d + 2642 \, a^{2} c^{2} d^{2} + 812 \, a^{2} c d^{3} + 419 \, a^{2} d^{4}\right )} \cos \left (f x + e\right ) - {\left (672 \, a^{2} c^{4} - 2304 \, a^{2} c^{3} d + 3008 \, a^{2} c^{2} d^{2} - 1792 \, a^{2} c d^{3} + 416 \, a^{2} d^{4} + 8 \, {\left (a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 73 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{4} + 4 \, {\left (9 \, a^{2} c^{3} d + 91 \, a^{2} c^{2} d^{2} + 667 \, a^{2} c d^{3} + 73 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (21 \, a^{2} c^{4} + 204 \, a^{2} c^{3} d + 1478 \, a^{2} c^{2} d^{2} - 388 \, a^{2} c d^{3} + 365 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (147 \, a^{2} c^{4} + 1836 \, a^{2} c^{3} d + 1138 \, a^{2} c^{2} d^{2} + 1708 \, a^{2} c d^{3} + 211 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{315 \, {\left ({\left (c^{5} d^{5} + 5 \, c^{4} d^{6} + 10 \, c^{3} d^{7} + 10 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{6} - 5 \, {\left (c^{6} d^{4} + 5 \, c^{5} d^{5} + 10 \, c^{4} d^{6} + 10 \, c^{3} d^{7} + 5 \, c^{2} d^{8} + c d^{9}\right )} f \cos \left (f x + e\right )^{5} - {\left (10 \, c^{7} d^{3} + 55 \, c^{6} d^{4} + 128 \, c^{5} d^{5} + 165 \, c^{4} d^{6} + 130 \, c^{3} d^{7} + 65 \, c^{2} d^{8} + 20 \, c d^{9} + 3 \, d^{10}\right )} f \cos \left (f x + e\right )^{4} + 10 \, {\left (c^{8} d^{2} + 5 \, c^{7} d^{3} + 11 \, c^{6} d^{4} + 15 \, c^{5} d^{5} + 15 \, c^{4} d^{6} + 11 \, c^{3} d^{7} + 5 \, c^{2} d^{8} + c d^{9}\right )} f \cos \left (f x + e\right )^{3} + {\left (5 \, c^{9} d + 35 \, c^{8} d^{2} + 120 \, c^{7} d^{3} + 260 \, c^{6} d^{4} + 378 \, c^{5} d^{5} + 370 \, c^{4} d^{6} + 240 \, c^{3} d^{7} + 100 \, c^{2} d^{8} + 25 \, c d^{9} + 3 \, d^{10}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{10} + 5 \, c^{9} d + 20 \, c^{8} d^{2} + 60 \, c^{7} d^{3} + 110 \, c^{6} d^{4} + 126 \, c^{5} d^{5} + 100 \, c^{4} d^{6} + 60 \, c^{3} d^{7} + 25 \, c^{2} d^{8} + 5 \, c d^{9}\right )} f \cos \left (f x + e\right ) - {\left (c^{10} + 10 \, c^{9} d + 45 \, c^{8} d^{2} + 120 \, c^{7} d^{3} + 210 \, c^{6} d^{4} + 252 \, c^{5} d^{5} + 210 \, c^{4} d^{6} + 120 \, c^{3} d^{7} + 45 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right )} f - {\left ({\left (c^{5} d^{5} + 5 \, c^{4} d^{6} + 10 \, c^{3} d^{7} + 10 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{5} + {\left (5 \, c^{6} d^{4} + 26 \, c^{5} d^{5} + 55 \, c^{4} d^{6} + 60 \, c^{3} d^{7} + 35 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (5 \, c^{7} d^{3} + 25 \, c^{6} d^{4} + 51 \, c^{5} d^{5} + 55 \, c^{4} d^{6} + 35 \, c^{3} d^{7} + 15 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, c^{8} d^{2} + 30 \, c^{7} d^{3} + 80 \, c^{6} d^{4} + 126 \, c^{5} d^{5} + 130 \, c^{4} d^{6} + 90 \, c^{3} d^{7} + 40 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{2} + {\left (5 \, c^{9} d + 25 \, c^{8} d^{2} + 60 \, c^{7} d^{3} + 100 \, c^{6} d^{4} + 126 \, c^{5} d^{5} + 110 \, c^{4} d^{6} + 60 \, c^{3} d^{7} + 20 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right ) + {\left (c^{10} + 10 \, c^{9} d + 45 \, c^{8} d^{2} + 120 \, c^{7} d^{3} + 210 \, c^{6} d^{4} + 252 \, c^{5} d^{5} + 210 \, c^{4} d^{6} + 120 \, c^{3} d^{7} + 45 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right )} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm="fricas")

[Out]

2/315*(672*a^2*c^4 - 2304*a^2*c^3*d + 3008*a^2*c^2*d^2 - 1792*a^2*c*d^3 + 416*a^2*d^4 + 8*(a^2*c^2*d^2 + 10*a^
2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^5 - 4*(9*a^2*c^3*d + 89*a^2*c^2*d^2 + 647*a^2*c*d^3 - 73*a^2*d^4)*cos(f*x +
e)^4 - (63*a^2*c^4 + 648*a^2*c^3*d + 4798*a^2*c^2*d^2 + 1504*a^2*c*d^3 + 1387*a^2*d^4)*cos(f*x + e)^3 + (231*
a^2*c^4 + 3060*a^2*c^3*d - 2158*a^2*c^2*d^2 + 4580*a^2*c*d^3 - 673*a^2*d^4)*cos(f*x + e)^2 + 2*(483*a^2*c^4 +
684*a^2*c^3*d + 2642*a^2*c^2*d^2 + 812*a^2*c*d^3 + 419*a^2*d^4)*cos(f*x + e) - (672*a^2*c^4 - 2304*a^2*c^3*d +
3008*a^2*c^2*d^2 - 1792*a^2*c*d^3 + 416*a^2*d^4 + 8*(a^2*c^2*d^2 + 10*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^4
+ 4*(9*a^2*c^3*d + 91*a^2*c^2*d^2 + 667*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^3 - 3*(21*a^2*c^4 + 204*a^2*c^3*d
+ 1478*a^2*c^2*d^2 - 388*a^2*c*d^3 + 365*a^2*d^4)*cos(f*x + e)^2 - 2*(147*a^2*c^4 + 1836*a^2*c^3*d + 1138*a^2
*c^2*d^2 + 1708*a^2*c*d^3 + 211*a^2*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x +
e) + c)/((c^5*d^5 + 5*c^4*d^6 + 10*c^3*d^7 + 10*c^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^6 - 5*(c^6*d^4 + 5*c
^5*d^5 + 10*c^4*d^6 + 10*c^3*d^7 + 5*c^2*d^8 + c*d^9)*f*cos(f*x + e)^5 - (10*c^7*d^3 + 55*c^6*d^4 + 128*c^5*d^
5 + 165*c^4*d^6 + 130*c^3*d^7 + 65*c^2*d^8 + 20*c*d^9 + 3*d^10)*f*cos(f*x + e)^4 + 10*(c^8*d^2 + 5*c^7*d^3 + 1
1*c^6*d^4 + 15*c^5*d^5 + 15*c^4*d^6 + 11*c^3*d^7 + 5*c^2*d^8 + c*d^9)*f*cos(f*x + e)^3 + (5*c^9*d + 35*c^8*d^2
+ 120*c^7*d^3 + 260*c^6*d^4 + 378*c^5*d^5 + 370*c^4*d^6 + 240*c^3*d^7 + 100*c^2*d^8 + 25*c*d^9 + 3*d^10)*f*co
s(f*x + e)^2 - (c^10 + 5*c^9*d + 20*c^8*d^2 + 60*c^7*d^3 + 110*c^6*d^4 + 126*c^5*d^5 + 100*c^4*d^6 + 60*c^3*d^
7 + 25*c^2*d^8 + 5*c*d^9)*f*cos(f*x + e) - (c^10 + 10*c^9*d + 45*c^8*d^2 + 120*c^7*d^3 + 210*c^6*d^4 + 252*c^5
*d^5 + 210*c^4*d^6 + 120*c^3*d^7 + 45*c^2*d^8 + 10*c*d^9 + d^10)*f - ((c^5*d^5 + 5*c^4*d^6 + 10*c^3*d^7 + 10*c
^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^5 + (5*c^6*d^4 + 26*c^5*d^5 + 55*c^4*d^6 + 60*c^3*d^7 + 35*c^2*d^8 + 1
0*c*d^9 + d^10)*f*cos(f*x + e)^4 - 2*(5*c^7*d^3 + 25*c^6*d^4 + 51*c^5*d^5 + 55*c^4*d^6 + 35*c^3*d^7 + 15*c^2*d
^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^3 - 2*(5*c^8*d^2 + 30*c^7*d^3 + 80*c^6*d^4 + 126*c^5*d^5 + 130*c^4*d^6 + 9
0*c^3*d^7 + 40*c^2*d^8 + 10*c*d^9 + d^10)*f*cos(f*x + e)^2 + (5*c^9*d + 25*c^8*d^2 + 60*c^7*d^3 + 100*c^6*d^4
+ 126*c^5*d^5 + 110*c^4*d^6 + 60*c^3*d^7 + 20*c^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e) + (c^10 + 10*c^9*d + 45
*c^8*d^2 + 120*c^7*d^3 + 210*c^6*d^4 + 252*c^5*d^5 + 210*c^4*d^6 + 120*c^3*d^7 + 45*c^2*d^8 + 10*c*d^9 + d^10)
*f)*sin(f*x + e))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(11/2),x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 26.14, size = 1155, normalized size = 3.64 \begin {gather*} -\frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {32\,a^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,1{}\mathrm {i}+c\,d\,10{}\mathrm {i}+d^2\,73{}\mathrm {i}\right )}{315\,d^3\,f\,{\left (c+d\right )}^5}-\frac {32\,a^2\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2+10\,c\,d+73\,d^2\right )}{315\,d^3\,f\,{\left (c+d\right )}^5}-\frac {32\,a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (25\,c^4-25\,c^3\,d+57\,c^2\,d^2-15\,c\,d^3+6\,d^4\right )}{5\,d^5\,f\,{\left (c+d\right )}^5}+\frac {32\,a^2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^4\,25{}\mathrm {i}-c^3\,d\,25{}\mathrm {i}+c^2\,d^2\,57{}\mathrm {i}-c\,d^3\,15{}\mathrm {i}+d^4\,6{}\mathrm {i}\right )}{5\,d^5\,f\,{\left (c+d\right )}^5}-\frac {16\,a^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (25\,c^4+318\,c^3\,d-20\,c^2\,d^2+194\,c\,d^3-5\,d^4\right )}{15\,d^5\,f\,{\left (c+d\right )}^5}+\frac {16\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^4\,25{}\mathrm {i}+c^3\,d\,318{}\mathrm {i}-c^2\,d^2\,20{}\mathrm {i}+c\,d^3\,194{}\mathrm {i}-d^4\,5{}\mathrm {i}\right )}{15\,d^5\,f\,{\left (c+d\right )}^5}+\frac {16\,a^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (7\,c^4+70\,c^3\,d+512\,c^2\,d^2+10\,c\,d^3+73\,d^4\right )}{35\,d^5\,f\,{\left (c+d\right )}^5}-\frac {16\,a^2\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^4\,7{}\mathrm {i}+c^3\,d\,70{}\mathrm {i}+c^2\,d^2\,512{}\mathrm {i}+c\,d^3\,10{}\mathrm {i}+d^4\,73{}\mathrm {i}\right )}{35\,d^5\,f\,{\left (c+d\right )}^5}+\frac {32\,a^2\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2+10\,c\,d+73\,d^2\right )}{35\,d^4\,f\,{\left (c+d\right )}^5}-\frac {32\,a^2\,c\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,1{}\mathrm {i}+c\,d\,10{}\mathrm {i}+d^2\,73{}\mathrm {i}\right )}{35\,d^4\,f\,{\left (c+d\right )}^5}\right )}{{\mathrm {e}}^{e\,11{}\mathrm {i}+f\,x\,11{}\mathrm {i}}-\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5}{{\left (c+d\right )}^5}+\frac {10\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\left (8\,c^4+8\,c^3\,d+12\,c^2\,d^2+4\,c\,d^3+d^4\right )}{d^4}+\frac {5\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (16\,c^3+8\,c^2\,d+8\,c\,d^2+d^3\right )}{d^3}-\frac {5\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\left (8\,c^2+2\,c\,d+d^2\right )}{d^2}-\frac {2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (16\,c^5+40\,c^4\,d+80\,c^3\,d^2+60\,c^2\,d^3+30\,c\,d^4+5\,d^5\right )}{d^5}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (10\,c+d\right )}{d}-\frac {5\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5\,\left (16\,c^3+8\,c^2\,d+8\,c\,d^2+d^3\right )}{d^3\,{\left (c+d\right )}^5}+\frac {5\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5\,\left (8\,c^2+2\,c\,d+d^2\right )}{d^2\,{\left (c+d\right )}^5}+\frac {2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5\,\left (16\,c^5+40\,c^4\,d+80\,c^3\,d^2+60\,c^2\,d^3+30\,c\,d^4+5\,d^5\right )}{d^5\,{\left (c+d\right )}^5}+\frac {{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\left (10\,c+d\right )\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5}{d\,{\left (c+d\right )}^5}-\frac {10\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5\,\left (8\,c^4+8\,c^3\,d+12\,c^2\,d^2+4\,c\,d^3+d^4\right )}{d^4\,{\left (c+d\right )}^5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(11/2),x)

[Out]

-((c + d*sin(e + f*x))^(1/2)*((32*a^2*exp(e*1i + f*x*1i)*(a + a*sin(e + f*x))^(1/2)*(c*d*10i + c^2*1i + d^2*73
i))/(315*d^3*f*(c + d)^5) - (32*a^2*exp(e*10i + f*x*10i)*(a + a*sin(e + f*x))^(1/2)*(10*c*d + c^2 + 73*d^2))/(
315*d^3*f*(c + d)^5) - (32*a^2*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2)*(25*c^4 - 25*c^3*d - 15*c*d^3 + 6
*d^4 + 57*c^2*d^2))/(5*d^5*f*(c + d)^5) + (32*a^2*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*(c^4*25i - c^3
*d*25i - c*d^3*15i + d^4*6i + c^2*d^2*57i))/(5*d^5*f*(c + d)^5) - (16*a^2*exp(e*4i + f*x*4i)*(a + a*sin(e + f*
x))^(1/2)*(194*c*d^3 + 318*c^3*d + 25*c^4 - 5*d^4 - 20*c^2*d^2))/(15*d^5*f*(c + d)^5) + (16*a^2*exp(e*7i + f*x
*7i)*(a + a*sin(e + f*x))^(1/2)*(c*d^3*194i + c^3*d*318i + c^4*25i - d^4*5i - c^2*d^2*20i))/(15*d^5*f*(c + d)^
5) + (16*a^2*exp(e*8i + f*x*8i)*(a + a*sin(e + f*x))^(1/2)*(10*c*d^3 + 70*c^3*d + 7*c^4 + 73*d^4 + 512*c^2*d^2
))/(35*d^5*f*(c + d)^5) - (16*a^2*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^(1/2)*(c*d^3*10i + c^3*d*70i + c^4*7
i + d^4*73i + c^2*d^2*512i))/(35*d^5*f*(c + d)^5) + (32*a^2*c*exp(e*2i + f*x*2i)*(a + a*sin(e + f*x))^(1/2)*(1
0*c*d + c^2 + 73*d^2))/(35*d^4*f*(c + d)^5) - (32*a^2*c*exp(e*9i + f*x*9i)*(a + a*sin(e + f*x))^(1/2)*(c*d*10i
+ c^2*1i + d^2*73i))/(35*d^4*f*(c + d)^5)))/(exp(e*11i + f*x*11i) - (c*1i + d*1i)^5/(c + d)^5 + (10*exp(e*7i
+ f*x*7i)*(4*c*d^3 + 8*c^3*d + 8*c^4 + d^4 + 12*c^2*d^2))/d^4 + (5*exp(e*3i + f*x*3i)*(8*c*d^2 + 8*c^2*d + 16*
c^3 + d^3))/d^3 - (5*exp(e*9i + f*x*9i)*(2*c*d + 8*c^2 + d^2))/d^2 - (2*exp(e*5i + f*x*5i)*(30*c*d^4 + 40*c^4*
d + 16*c^5 + 5*d^5 + 60*c^2*d^3 + 80*c^3*d^2))/d^5 - (exp(e*1i + f*x*1i)*(10*c + d))/d - (5*exp(e*8i + f*x*8i)
*(c*1i + d*1i)^5*(8*c*d^2 + 8*c^2*d + 16*c^3 + d^3))/(d^3*(c + d)^5) + (5*exp(e*2i + f*x*2i)*(c*1i + d*1i)^5*(
2*c*d + 8*c^2 + d^2))/(d^2*(c + d)^5) + (2*exp(e*6i + f*x*6i)*(c*1i + d*1i)^5*(30*c*d^4 + 40*c^4*d + 16*c^5 +
5*d^5 + 60*c^2*d^3 + 80*c^3*d^2))/(d^5*(c + d)^5) + (exp(e*10i + f*x*10i)*(10*c + d)*(c*1i + d*1i)^5)/(d*(c +
d)^5) - (10*exp(e*4i + f*x*4i)*(c*1i + d*1i)^5*(4*c*d^3 + 8*c^3*d + 8*c^4 + d^4 + 12*c^2*d^2))/(d^4*(c + d)^5)
)

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