3.6.86 \(\int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{9/2}} \, dx\) [586]

Optimal. Leaf size=254 \[ \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {4 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]

[Out]

6/35*a^3*(c-d)*(c+5*d)*cos(f*x+e)/d^2/(c+d)^2/f/(c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)-2/105*a^3*(3*c^2
+22*c*d+115*d^2)*cos(f*x+e)/d^2/(c+d)^3/f/(c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)+2/7*a^2*(c-d)*cos(f*x+
e)*(a+a*sin(f*x+e))^(1/2)/d/(c+d)/f/(c+d*sin(f*x+e))^(7/2)-4/105*a^3*(3*c^2+22*c*d+115*d^2)*cos(f*x+e)/d^2/(c+
d)^4/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.41, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {4 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 f (c+d)^4 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (3 c^2+22 c d+115 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))^{3/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{7 d f (c+d) (c+d \sin (e+f x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(9/2),x]

[Out]

(2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(7*d*(c + d)*f*(c + d*Sin[e + f*x])^(7/2)) + (6*a^3*(c -
d)*(c + 5*d)*Cos[e + f*x])/(35*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (2*a^3*
(3*c^2 + 22*c*d + 115*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])^(3
/2)) - (4*a^3*(3*c^2 + 22*c*d + 115*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^4*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))^{9/2}} \, dx &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}-\frac {(2 a) \int \frac {\sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a (c-15 d)-\frac {1}{2} a (3 c+11 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{7/2}} \, dx}{7 d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac {\left (a^2 \left (3 c^2+22 c d+115 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{35 d^2 (c+d)^2}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^3 \left (3 c^2+22 c d+115 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))^{3/2}}+\frac {\left (2 a^2 \left (3 c^2+22 c d+115 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{105 d^2 (c+d)^3}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{7 d (c+d) f (c+d \sin (e+f x))^{7/2}}+\frac {6 a^3 (c-d) (c+5 d) \cos (e+f x)}{35 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {2 a^3 \left (3 c^2+22 c d+115 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))^{3/2}}-\frac {4 a^3 \left (3 c^2+22 c d+115 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 2.73, size = 216, normalized size = 0.85 \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 (-623 c^3-495 c^2 d-977 c d^2-145 d^3+\left (21 c^3+157 c^2 d+827 c d^2+115 d^3\right ) \cos (2 (e+f x))-\left (196 c^3+1865 c^2 d+694 c d^2+465 d^3\right ) \sin (e+f x)+3 c^2 d \sin (3 (e+f x))+22 c d^2 \sin (3 (e+f x))+115 d^3 \sin (3 (e+f x))\right )}{105 (c+d)^4 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))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(9/2),x]

[Out]

(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(-623*c^3 - 495*c^2*d - 977*c*d^2 - 145*
d^3 + (21*c^3 + 157*c^2*d + 827*c*d^2 + 115*d^3)*Cos[2*(e + f*x)] - (196*c^3 + 1865*c^2*d + 694*c*d^2 + 465*d^
3)*Sin[e + f*x] + 3*c^2*d*Sin[3*(e + f*x)] + 22*c*d^2*Sin[3*(e + f*x)] + 115*d^3*Sin[3*(e + f*x)]))/(105*(c +
d)^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x])^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1222\) vs. \(2(230)=460\).
time = 7.29, size = 1223, normalized size = 4.81

method	result	size
default	\(\text {Expression too large to display}\)	\(1223\)

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))^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/105/f*(a*(1+sin(f*x+e)))^(5/2)*(c+d*sin(f*x+e))^(1/2)*(6*cos(f*x+e)^10*c^2*d^5-2176*c^5*d^2+1664*c^3*d^4-38
4*c*d^6+368*cos(f*x+e)^2*sin(f*x+e)*c^6*d-3344*cos(f*x+e)^2*sin(f*x+e)*c^5*d^2-5008*cos(f*x+e)^2*sin(f*x+e)*c^
4*d^3+4560*cos(f*x+e)^2*sin(f*x+e)*c^3*d^4-640*d^7+44*cos(f*x+e)^10*c*d^6+575*cos(f*x+e)^8*sin(f*x+e)*d^7+48*c
os(f*x+e)^8*c^4*d^3+257*cos(f*x+e)^8*c^3*d^4+895*cos(f*x+e)^8*c^2*d^5-485*cos(f*x+e)^8*c*d^6-2350*cos(f*x+e)^6
*sin(f*x+e)*d^7-78*cos(f*x+e)^6*c^6*d-302*cos(f*x+e)^6*c^5*d^2-172*cos(f*x+e)^6*c^4*d^3-2968*cos(f*x+e)^6*c^3*
d^4-5370*cos(f*x+e)^6*c^2*d^5+1590*cos(f*x+e)^6*c*d^6-21*cos(f*x+e)^4*sin(f*x+e)*c^7+3615*cos(f*x+e)^4*sin(f*x
+e)*d^7+39*cos(f*x+e)^4*c^6*d-1879*cos(f*x+e)^4*c^5*d^2-3397*cos(f*x+e)^4*c^4*d^3+6439*cos(f*x+e)^4*c^3*d^4+10
373*cos(f*x+e)^4*c^2*d^5-2285*cos(f*x+e)^4*c*d^6+112*cos(f*x+e)^2*sin(f*x+e)*c^7-2480*cos(f*x+e)^2*sin(f*x+e)*
d^7-944*cos(f*x+e)^2*c^6*d+4432*cos(f*x+e)^2*c^5*d^2+6480*cos(f*x+e)^2*c^4*d^3-5392*cos(f*x+e)^2*c^3*d^4-8336*
cos(f*x+e)^2*c^2*d^5+1520*cos(f*x+e)^2*c*d^6-1152*sin(f*x+e)*c^6*d+2176*sin(f*x+e)*c^5*d^2+2944*sin(f*x+e)*c^4
*d^3-1664*sin(f*x+e)*c^3*d^4-2432*sin(f*x+e)*c^2*d^5+384*sin(f*x+e)*c*d^6-560*cos(f*x+e)^2*c^7+2800*cos(f*x+e)
^2*d^7-896*sin(f*x+e)*c^7+640*sin(f*x+e)*d^7+230*cos(f*x+e)^10*d^7-1555*cos(f*x+e)^8*d^7+3940*cos(f*x+e)^6*d^7
-35*cos(f*x+e)^4*c^7-4775*cos(f*x+e)^4*d^7+896*c^7+7120*cos(f*x+e)^2*sin(f*x+e)*c^2*d^5-1328*cos(f*x+e)^2*sin(
f*x+e)*c*d^6+cos(f*x+e)^4*sin(f*x+e)*c^6*d+479*cos(f*x+e)^4*sin(f*x+e)*c^5*d^2+1261*cos(f*x+e)^4*sin(f*x+e)*c^
4*d^3-4367*cos(f*x+e)^4*sin(f*x+e)*c^3*d^4-7117*cos(f*x+e)^4*sin(f*x+e)*c^2*d^5+1669*cos(f*x+e)^4*sin(f*x+e)*c
*d^6+3*cos(f*x+e)^8*sin(f*x+e)*c^3*d^4+37*cos(f*x+e)^8*sin(f*x+e)*c^2*d^5+225*cos(f*x+e)^8*sin(f*x+e)*c*d^6+10
2*cos(f*x+e)^6*sin(f*x+e)*c^5*d^2+518*cos(f*x+e)^6*sin(f*x+e)*c^4*d^3+1408*cos(f*x+e)^6*sin(f*x+e)*c^3*d^4+239
2*cos(f*x+e)^6*sin(f*x+e)*c^2*d^5-950*cos(f*x+e)^6*sin(f*x+e)*c*d^6-2944*c^4*d^3+2432*c^2*d^5+1152*c^6*d)/cos(
f*x+e)^5/(cos(f*x+e)^2*d^2+c^2-d^2)^4/(c+d)^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (242) = 484\).
time = 0.74, size = 731, normalized size = 2.88 \begin {gather*} -\frac {2 \, {\left ({\left (301 \, c^{4} + 169 \, c^{3} d + 75 \, c^{2} d^{2} + 15 \, c d^{3}\right )} a^{\frac {5}{2}} - \frac {3 \, {\left (35 \, c^{4} - 763 \, c^{3} d - 297 \, c^{2} d^{2} - 85 \, c d^{3} - 10 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {6 \, {\left (182 \, c^{4} - 127 \, c^{3} d + 1059 \, c^{2} d^{2} + 251 \, c d^{3} + 35 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {14 \, {\left (50 \, c^{4} - 421 \, c^{3} d + 201 \, c^{2} d^{2} - 535 \, c d^{3} - 55 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, {\left (11 \, c^{4} - 36 \, c^{3} d + 80 \, c^{2} d^{2} - 40 \, c d^{3} + 25 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {126 \, {\left (11 \, c^{4} - 36 \, c^{3} d + 80 \, c^{2} d^{2} - 40 \, c d^{3} + 25 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {14 \, {\left (50 \, c^{4} - 421 \, c^{3} d + 201 \, c^{2} d^{2} - 535 \, c d^{3} - 55 \, d^{4}\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 (182 \, c^{4} - 127 \, c^{3} d + 1059 \, c^{2} d^{2} + 251 \, c d^{3} + 35 \, d^{4}\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 (35 \, c^{4} - 763 \, c^{3} d - 297 \, c^{2} d^{2} - 85 \, c d^{3} - 10 \, d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {{\left (301 \, c^{4} + 169 \, c^{3} d + 75 \, c^{2} d^{2} + 15 \, c d^{3}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{2}}{105 \, {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} + \frac {2 \, {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\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 {9}{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))^(9/2),x, algorithm="maxima")

[Out]

-2/105*((301*c^4 + 169*c^3*d + 75*c^2*d^2 + 15*c*d^3)*a^(5/2) - 3*(35*c^4 - 763*c^3*d - 297*c^2*d^2 - 85*c*d^3
- 10*d^4)*a^(5/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 6*(182*c^4 - 127*c^3*d + 1059*c^2*d^2 + 251*c*d^3 + 35*d^
4)*a^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 14*(50*c^4 - 421*c^3*d + 201*c^2*d^2 - 535*c*d^3 - 55*d^4)*a^
(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*(11*c^4 - 36*c^3*d + 80*c^2*d^2 - 40*c*d^3 + 25*d^4)*a^(5/2)*s
in(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*(11*c^4 - 36*c^3*d + 80*c^2*d^2 - 40*c*d^3 + 25*d^4)*a^(5/2)*sin(f*x
+ e)^5/(cos(f*x + e) + 1)^5 + 14*(50*c^4 - 421*c^3*d + 201*c^2*d^2 - 535*c*d^3 - 55*d^4)*a^(5/2)*sin(f*x + e)^
6/(cos(f*x + e) + 1)^6 - 6*(182*c^4 - 127*c^3*d + 1059*c^2*d^2 + 251*c*d^3 + 35*d^4)*a^(5/2)*sin(f*x + e)^7/(c
os(f*x + e) + 1)^7 + 3*(35*c^4 - 763*c^3*d - 297*c^2*d^2 - 85*c*d^3 - 10*d^4)*a^(5/2)*sin(f*x + e)^8/(cos(f*x
+ e) + 1)^8 - (301*c^4 + 169*c^3*d + 75*c^2*d^2 + 15*c*d^3)*a^(5/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9)*(sin(
f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^2/((c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 + 2*(c^4 + 4*c^3*d + 6*c^2*
d^2 + 4*c*d^3 + d^4)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*sin(f*x
+ e)^4/(cos(f*x + e) + 1)^4)*(c + 2*d*sin(f*x + e)/(cos(f*x + e) + 1) + c*sin(f*x + e)^2/(cos(f*x + e) + 1)^2
)^(9/2)*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1053 vs. \(2 (242) = 484\).
time = 0.41, size = 1053, normalized size = 4.15 \begin {gather*} -\frac {2 \, {\left (224 \, a^{2} c^{3} - 608 \, a^{2} c^{2} d + 544 \, a^{2} c d^{2} - 160 \, a^{2} d^{3} - 2 \, {\left (3 \, a^{2} c^{2} d + 22 \, a^{2} c d^{2} + 115 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} - {\left (21 \, a^{2} c^{3} + 157 \, a^{2} c^{2} d + 827 \, a^{2} c d^{2} + 115 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (77 \, a^{2} c^{3} + 783 \, a^{2} c^{2} d - 425 \, a^{2} c d^{2} + 405 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (161 \, a^{2} c^{3} + 163 \, a^{2} c^{2} d + 451 \, a^{2} c d^{2} + 65 \, a^{2} d^{3}\right )} \cos \left (f x + e\right ) - {\left (224 \, a^{2} c^{3} - 608 \, a^{2} c^{2} d + 544 \, a^{2} c d^{2} - 160 \, a^{2} d^{3} + 2 \, {\left (3 \, a^{2} c^{2} d + 22 \, a^{2} c d^{2} + 115 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (21 \, a^{2} c^{3} + 151 \, a^{2} c^{2} d + 783 \, a^{2} c d^{2} - 115 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (49 \, a^{2} c^{3} + 467 \, a^{2} c^{2} d + 179 \, a^{2} c d^{2} + 145 \, a^{2} d^{3}\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}}{105 \, {\left ({\left (c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{5} + {\left (4 \, c^{5} d^{3} + 17 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 22 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, c^{6} d^{2} + 12 \, c^{5} d^{3} + 19 \, c^{4} d^{4} + 16 \, c^{3} d^{5} + 9 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{3} - 2 \, {\left (2 \, c^{7} d + 11 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 43 \, c^{4} d^{4} + 42 \, c^{3} d^{5} + 25 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{2} + {\left (c^{8} + 4 \, c^{7} d + 12 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 38 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 12 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right ) + {\left (c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right )} f + {\left ({\left (c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{4} - 4 \, {\left (c^{5} d^{3} + 4 \, c^{4} d^{4} + 6 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right )} f \cos \left (f x + e\right )^{3} - 2 \, {\left (3 \, c^{6} d^{2} + 14 \, c^{5} d^{3} + 27 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 17 \, c^{2} d^{6} + 6 \, c d^{7} + d^{8}\right )} f \cos \left (f x + e\right )^{2} + 4 \, {\left (c^{7} d + 4 \, c^{6} d^{2} + 7 \, c^{5} d^{3} + 8 \, c^{4} d^{4} + 7 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\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))^(9/2),x, algorithm="fricas")

[Out]

-2/105*(224*a^2*c^3 - 608*a^2*c^2*d + 544*a^2*c*d^2 - 160*a^2*d^3 - 2*(3*a^2*c^2*d + 22*a^2*c*d^2 + 115*a^2*d^
3)*cos(f*x + e)^4 - (21*a^2*c^3 + 157*a^2*c^2*d + 827*a^2*c*d^2 + 115*a^2*d^3)*cos(f*x + e)^3 + (77*a^2*c^3 +
783*a^2*c^2*d - 425*a^2*c*d^2 + 405*a^2*d^3)*cos(f*x + e)^2 + 2*(161*a^2*c^3 + 163*a^2*c^2*d + 451*a^2*c*d^2 +
65*a^2*d^3)*cos(f*x + e) - (224*a^2*c^3 - 608*a^2*c^2*d + 544*a^2*c*d^2 - 160*a^2*d^3 + 2*(3*a^2*c^2*d + 22*a
^2*c*d^2 + 115*a^2*d^3)*cos(f*x + e)^3 - (21*a^2*c^3 + 151*a^2*c^2*d + 783*a^2*c*d^2 - 115*a^2*d^3)*cos(f*x +
e)^2 - 2*(49*a^2*c^3 + 467*a^2*c^2*d + 179*a^2*c*d^2 + 145*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x
+ e) + a)*sqrt(d*sin(f*x + e) + c)/((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^5 + (4*c
^5*d^3 + 17*c^4*d^4 + 28*c^3*d^5 + 22*c^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^4 - 2*(3*c^6*d^2 + 12*c^5*d^3 +
19*c^4*d^4 + 16*c^3*d^5 + 9*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^3 - 2*(2*c^7*d + 11*c^6*d^2 + 28*c^5*d^3 +
43*c^4*d^4 + 42*c^3*d^5 + 25*c^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^2 + (c^8 + 4*c^7*d + 12*c^6*d^2 + 28*c^5
*d^3 + 38*c^4*d^4 + 28*c^3*d^5 + 12*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e) + (c^8 + 8*c^7*d + 28*c^6*d^2 + 56
*c^5*d^3 + 70*c^4*d^4 + 56*c^3*d^5 + 28*c^2*d^6 + 8*c*d^7 + d^8)*f + ((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d
^7 + d^8)*f*cos(f*x + e)^4 - 4*(c^5*d^3 + 4*c^4*d^4 + 6*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f*x + e)^3 - 2*(3*c
^6*d^2 + 14*c^5*d^3 + 27*c^4*d^4 + 28*c^3*d^5 + 17*c^2*d^6 + 6*c*d^7 + d^8)*f*cos(f*x + e)^2 + 4*(c^7*d + 4*c^
6*d^2 + 7*c^5*d^3 + 8*c^4*d^4 + 7*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f*x + e) + (c^8 + 8*c^7*d + 28*c^6*d^2 +
56*c^5*d^3 + 70*c^4*d^4 + 56*c^3*d^5 + 28*c^2*d^6 + 8*c*d^7 + d^8)*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))**(9/2),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))^(9/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 20.61, size = 862, normalized size = 3.39 \begin {gather*} \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {8\,a^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (3\,c^2+22\,c\,d+115\,d^2\right )}{105\,d^3\,f\,{\left (c+d\right )}^4}+\frac {8\,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\,3{}\mathrm {i}+c\,d\,22{}\mathrm {i}+d^2\,115{}\mathrm {i}\right )}{105\,d^3\,f\,{\left (c+d\right )}^4}-\frac {8\,a^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (30\,c^3-25\,c^2\,d+36\,c\,d^2-5\,d^3\right )}{3\,d^4\,f\,{\left (c+d\right )}^4}-\frac {8\,a^2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^3\,30{}\mathrm {i}-c^2\,d\,25{}\mathrm {i}+c\,d^2\,36{}\mathrm {i}-d^3\,5{}\mathrm {i}\right )}{3\,d^4\,f\,{\left (c+d\right )}^4}-\frac {8\,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^3+244\,c^2\,d-19\,c\,d^2+50\,d^3\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}-\frac {8\,a^2\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^3\,25{}\mathrm {i}+c^2\,d\,244{}\mathrm {i}-c\,d^2\,19{}\mathrm {i}+d^3\,50{}\mathrm {i}\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}+\frac {8\,a^2\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (3\,c^2+22\,c\,d+115\,d^2\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}+\frac {8\,a^2\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,3{}\mathrm {i}+c\,d\,22{}\mathrm {i}+d^2\,115{}\mathrm {i}\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}\right )}{{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}+\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{{\left (c+d\right )}^4}-\frac {4\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (8\,c^3+6\,c^2\,d+6\,c\,d^2+d^3\right )}{d^3}-\frac {4\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\left (6\,c^2+2\,c\,d+d^2\right )}{d^2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (8\,c+d\right )}{d}+\frac {2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (8\,c^4+16\,c^3\,d+24\,c^2\,d^2+12\,c\,d^3+3\,d^4\right )}{d^4}-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,\left (8\,c^3+6\,c^2\,d+6\,c\,d^2+d^3\right )\,4{}\mathrm {i}}{d^3\,{\left (c+d\right )}^4}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,\left (6\,c^2+2\,c\,d+d^2\right )\,4{}\mathrm {i}}{d^2\,{\left (c+d\right )}^4}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (8\,c+d\right )\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{d\,{\left (c+d\right )}^4}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^4\,\left (8\,c^4+16\,c^3\,d+24\,c^2\,d^2+12\,c\,d^3+3\,d^4\right )\,2{}\mathrm {i}}{d^4\,{\left (c+d\right )}^4}} \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))^(9/2),x)

[Out]

((c + d*sin(e + f*x))^(1/2)*((8*a^2*exp(e*8i + f*x*8i)*(a + a*sin(e + f*x))^(1/2)*(22*c*d + 3*c^2 + 115*d^2))/
(105*d^3*f*(c + d)^4) + (8*a^2*exp(e*1i + f*x*1i)*(a + a*sin(e + f*x))^(1/2)*(c*d*22i + c^2*3i + d^2*115i))/(1
05*d^3*f*(c + d)^4) - (8*a^2*exp(e*4i + f*x*4i)*(a + a*sin(e + f*x))^(1/2)*(36*c*d^2 - 25*c^2*d + 30*c^3 - 5*d
^3))/(3*d^4*f*(c + d)^4) - (8*a^2*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2*36i - c^2*d*25i + c^3*3
0i - d^3*5i))/(3*d^4*f*(c + d)^4) - (8*a^2*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2)*(244*c^2*d - 19*c*d^2
+ 25*c^3 + 50*d^3))/(15*d^4*f*(c + d)^4) - (8*a^2*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^(1/2)*(c^2*d*244i -
c*d^2*19i + c^3*25i + d^3*50i))/(15*d^4*f*(c + d)^4) + (8*a^2*c*exp(e*2i + f*x*2i)*(a + a*sin(e + f*x))^(1/2)
*(22*c*d + 3*c^2 + 115*d^2))/(15*d^4*f*(c + d)^4) + (8*a^2*c*exp(e*7i + f*x*7i)*(a + a*sin(e + f*x))^(1/2)*(c*
d*22i + c^2*3i + d^2*115i))/(15*d^4*f*(c + d)^4)))/(exp(e*9i + f*x*9i) + ((c*1i + d*1i)^4*1i)/(c + d)^4 - (4*e
xp(e*3i + f*x*3i)*(6*c*d^2 + 6*c^2*d + 8*c^3 + d^3))/d^3 - (4*exp(e*7i + f*x*7i)*(2*c*d + 6*c^2 + d^2))/d^2 +
(exp(e*1i + f*x*1i)*(8*c + d))/d + (2*exp(e*5i + f*x*5i)*(12*c*d^3 + 16*c^3*d + 8*c^4 + 3*d^4 + 24*c^2*d^2))/d
^4 - (exp(e*6i + f*x*6i)*(c*1i + d*1i)^4*(6*c*d^2 + 6*c^2*d + 8*c^3 + d^3)*4i)/(d^3*(c + d)^4) - (exp(e*2i + f
*x*2i)*(c*1i + d*1i)^4*(2*c*d + 6*c^2 + d^2)*4i)/(d^2*(c + d)^4) + (exp(e*8i + f*x*8i)*(8*c + d)*(c*1i + d*1i)
^4*1i)/(d*(c + d)^4) + (exp(e*4i + f*x*4i)*(c*1i + d*1i)^4*(12*c*d^3 + 16*c^3*d + 8*c^4 + 3*d^4 + 24*c^2*d^2)*
2i)/(d^4*(c + d)^4))

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