3.6.78 \(\int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx\) [578]

Optimal. Leaf size=229 \[ \frac {2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^2 (c+13 d) \cos (e+f x)}{35 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^2 (c+13 d) \cos (e+f x)}{105 d (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a^2 (c+13 d) \cos (e+f x)}{105 d (c+d)^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]

[Out]

2/7*a^2*(c-d)*cos(f*x+e)/d/(c+d)/f/(c+d*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(1/2)-2/35*a^2*(c+13*d)*cos(f*x+e)/
d/(c+d)^2/f/(c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)-8/105*a^2*(c+13*d)*cos(f*x+e)/d/(c+d)^3/f/(c+d*sin(f
*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-16/105*a^2*(c+13*d)*cos(f*x+e)/d/(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 = 0.29, antiderivative size = 229, 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, 21, 2851, 2850} \begin {gather*} -\frac {16 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^4 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {8 a^2 (c+13 d) \cos (e+f x)}{105 d f (c+d)^3 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac {2 a^2 (c+13 d) \cos (e+f x)}{35 d 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)}{7 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(9/2),x]

[Out]

(2*a^2*(c - d)*Cos[e + f*x])/(7*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a^2*(c +
13*d)*Cos[e + f*x])/(35*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a^2*(c + 13*d
)*Cos[e + f*x])/(105*d*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a^2*(c + 13*d)*C
os[e + f*x])/(105*d*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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]

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{9/2}} \, dx &=\frac {2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {(2 a) \int \frac {-\frac {1}{2} a (c+13 d)-\frac {1}{2} a (c+13 d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}} \, dx}{7 d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}+\frac {(a (c+13 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx}{7 d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^2 (c+13 d) \cos (e+f x)}{35 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac {(4 a (c+13 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{35 d (c+d)^2}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^2 (c+13 d) \cos (e+f x)}{35 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^2 (c+13 d) \cos (e+f x)}{105 d (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {(8 a (c+13 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{105 d (c+d)^3}\\ &=\frac {2 a^2 (c-d) \cos (e+f x)}{7 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^2 (c+13 d) \cos (e+f x)}{35 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^2 (c+13 d) \cos (e+f x)}{105 d (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a^2 (c+13 d) \cos (e+f x)}{105 d (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 = 1.02, size = 193, normalized size = 0.84 \begin {gather*} -\frac {2 a \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 (175 c^3+147 c^2 d+253 c d^2+41 d^3-2 d \left (7 c^2+92 c d+13 d^2\right ) \cos (2 (e+f x))+\left (35 c^3+469 c^2 d+191 c d^2+117 d^3\right ) \sin (e+f x)-2 c d^2 \sin (3 (e+f x))-26 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])^(3/2)/(c + d*Sin[e + f*x])^(9/2),x]

[Out]

(-2*a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(175*c^3 + 147*c^2*d + 253*c*d^2 + 41*d
^3 - 2*d*(7*c^2 + 92*c*d + 13*d^2)*Cos[2*(e + f*x)] + (35*c^3 + 469*c^2*d + 191*c*d^2 + 117*d^3)*Sin[e + f*x]
- 2*c*d^2*Sin[3*(e + f*x)] - 26*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. \(978\) vs. \(2(205)=410\).
time = 2.23, size = 979, normalized size = 4.28

method	result	size
default	\(\text {Expression too large to display}\)	\(979\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/105/f*(a*(1+sin(f*x+e)))^(3/2)*(c+d*sin(f*x+e))^(1/2)*(776*c^4*d^3*sin(f*x+e)-136*c^3*d^4*sin(f*x+e)-424*c^
2*d^5*sin(f*x+e)+120*c*d^6*sin(f*x+e)-776*c^4*d^3+280*c^7-296*c^5*d^2-152*d^7+424*c^2*d^5+152*d^7*sin(f*x+e)+1
36*c^3*d^4-120*c*d^6+504*c^6*d-4*cos(f*x+e)^8*c^2*d^5-64*cos(f*x+e)^8*c*d^6-455*sin(f*x+e)*cos(f*x+e)^6*d^7+4*
cos(f*x+e)^6*c^4*d^3-149*cos(f*x+e)^6*c^3*d^4-443*cos(f*x+e)^6*c^2*d^5+345*cos(f*x+e)^6*c*d^6+750*sin(f*x+e)*c
os(f*x+e)^4*d^7-112*cos(f*x+e)^4*c^6*d-670*cos(f*x+e)^4*c^5*d^2-1398*cos(f*x+e)^4*c^4*d^3+56*cos(f*x+e)^4*c^3*
d^4+1232*cos(f*x+e)^4*c^2*d^5-618*cos(f*x+e)^4*c*d^6-35*sin(f*x+e)*cos(f*x+e)^2*c^7-551*sin(f*x+e)*cos(f*x+e)^
2*d^7-259*cos(f*x+e)^2*c^6*d+1035*cos(f*x+e)^2*c^5*d^2+2185*cos(f*x+e)^2*c^4*d^3-43*cos(f*x+e)^2*c^3*d^4-1209*
cos(f*x+e)^2*c^2*d^5+457*cos(f*x+e)^2*c*d^6-504*sin(f*x+e)*c^6*d+296*sin(f*x+e)*c^5*d^2+104*sin(f*x+e)*cos(f*x
+e)^8*d^7+7*sin(f*x+e)*cos(f*x+e)^2*c^6*d-887*sin(f*x+e)*cos(f*x+e)^2*c^5*d^2-1797*sin(f*x+e)*cos(f*x+e)^2*c^4
*d^3-25*sin(f*x+e)*cos(f*x+e)^2*c^3*d^4+997*sin(f*x+e)*cos(f*x+e)^2*c^2*d^5-397*sin(f*x+e)*cos(f*x+e)^2*c*d^6+
8*sin(f*x+e)*cos(f*x+e)^8*c*d^6+29*sin(f*x+e)*cos(f*x+e)^6*c^3*d^4+371*sin(f*x+e)*cos(f*x+e)^6*c^2*d^5-113*sin
(f*x+e)*cos(f*x+e)^6*c*d^6+106*sin(f*x+e)*cos(f*x+e)^4*c^5*d^2+754*sin(f*x+e)*cos(f*x+e)^4*c^4*d^3+72*sin(f*x+
e)*cos(f*x+e)^4*c^3*d^4-944*sin(f*x+e)*cos(f*x+e)^4*c^2*d^5+382*sin(f*x+e)*cos(f*x+e)^4*c*d^6-156*cos(f*x+e)^8
*d^7+635*cos(f*x+e)^6*d^7-954*cos(f*x+e)^4*d^7-105*cos(f*x+e)^2*c^7+627*cos(f*x+e)^2*d^7-280*sin(f*x+e)*c^7)/c
os(f*x+e)^3/(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. 780 vs. \(2 (217) = 434\).
time = 0.64, size = 780, normalized size = 3.41 \begin {gather*} -\frac {2 \, {\left ({\left (175 \, c^{4} + 133 \, c^{3} d + 69 \, c^{2} d^{2} + 15 \, c d^{3}\right )} a^{\frac {3}{2}} - \frac {3 \, {\left (35 \, c^{4} - 385 \, c^{3} d - 189 \, c^{2} d^{2} - 67 \, c d^{3} - 10 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {18 \, {\left (35 \, c^{4} - 28 \, c^{3} d + 166 \, c^{2} d^{2} + 44 \, c d^{3} + 7 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {14 \, {\left (35 \, c^{4} - 220 \, c^{3} d + 102 \, c^{2} d^{2} - 244 \, c d^{3} - 25 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {42 \, {\left (20 \, c^{4} - 61 \, c^{3} d + 117 \, c^{2} d^{2} - 55 \, c d^{3} + 35 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {42 \, {\left (20 \, c^{4} - 61 \, c^{3} d + 117 \, c^{2} d^{2} - 55 \, c d^{3} + 35 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {14 \, {\left (35 \, c^{4} - 220 \, c^{3} d + 102 \, c^{2} d^{2} - 244 \, c d^{3} - 25 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {18 \, {\left (35 \, c^{4} - 28 \, c^{3} d + 166 \, c^{2} d^{2} + 44 \, c d^{3} + 7 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {3 \, {\left (35 \, c^{4} - 385 \, c^{3} d - 189 \, c^{2} d^{2} - 67 \, c d^{3} - 10 \, d^{4}\right )} a^{\frac {3}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {{\left (175 \, c^{4} + 133 \, c^{3} d + 69 \, c^{2} d^{2} + 15 \, c d^{3}\right )} a^{\frac {3}{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 )}^{3}}{105 \, {\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} + \frac {3 \, {\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 {3 \, {\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}} + \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 )^{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 {9}{2}} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

-2/105*((175*c^4 + 133*c^3*d + 69*c^2*d^2 + 15*c*d^3)*a^(3/2) - 3*(35*c^4 - 385*c^3*d - 189*c^2*d^2 - 67*c*d^3
- 10*d^4)*a^(3/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 18*(35*c^4 - 28*c^3*d + 166*c^2*d^2 + 44*c*d^3 + 7*d^4)*a
^(3/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 14*(35*c^4 - 220*c^3*d + 102*c^2*d^2 - 244*c*d^3 - 25*d^4)*a^(3/2
)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 42*(20*c^4 - 61*c^3*d + 117*c^2*d^2 - 55*c*d^3 + 35*d^4)*a^(3/2)*sin(f
*x + e)^4/(cos(f*x + e) + 1)^4 - 42*(20*c^4 - 61*c^3*d + 117*c^2*d^2 - 55*c*d^3 + 35*d^4)*a^(3/2)*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 + 14*(35*c^4 - 220*c^3*d + 102*c^2*d^2 - 244*c*d^3 - 25*d^4)*a^(3/2)*sin(f*x + e)^6/(c
os(f*x + e) + 1)^6 - 18*(35*c^4 - 28*c^3*d + 166*c^2*d^2 + 44*c*d^3 + 7*d^4)*a^(3/2)*sin(f*x + e)^7/(cos(f*x +
e) + 1)^7 + 3*(35*c^4 - 385*c^3*d - 189*c^2*d^2 - 67*c*d^3 - 10*d^4)*a^(3/2)*sin(f*x + e)^8/(cos(f*x + e) + 1
)^8 - (175*c^4 + 133*c^3*d + 69*c^2*d^2 + 15*c*d^3)*a^(3/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9)*(sin(f*x + e)
^2/(cos(f*x + e) + 1)^2 + 1)^3/((c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 + 3*(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 + 3*(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^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*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)^(9/2)*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (217) = 434\).
time = 0.40, size = 957, normalized size = 4.18 \begin {gather*} \frac {2 \, {\left (8 \, {\left (a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )^{4} - 140 \, a c^{3} + 308 \, a c^{2} d - 244 \, a c d^{2} + 76 \, a d^{3} + 4 \, {\left (7 \, a c^{2} d + 92 \, a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (35 \, a c^{3} + 441 \, a c^{2} d - 167 \, a c d^{2} + 195 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (175 \, a c^{3} + 161 \, a c^{2} d + 437 \, a c d^{2} + 67 \, a d^{3}\right )} \cos \left (f x + e\right ) + {\left (140 \, a c^{3} - 308 \, a c^{2} d + 244 \, a c d^{2} - 76 \, a d^{3} + 8 \, {\left (a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} - 4 \, {\left (7 \, a c^{2} d + 90 \, a c d^{2} - 13 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (35 \, a c^{3} + 469 \, a c^{2} d + 193 \, a c d^{2} + 143 \, a 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))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

2/105*(8*(a*c*d^2 + 13*a*d^3)*cos(f*x + e)^4 - 140*a*c^3 + 308*a*c^2*d - 244*a*c*d^2 + 76*a*d^3 + 4*(7*a*c^2*d
+ 92*a*c*d^2 + 13*a*d^3)*cos(f*x + e)^3 - (35*a*c^3 + 441*a*c^2*d - 167*a*c*d^2 + 195*a*d^3)*cos(f*x + e)^2 -
(175*a*c^3 + 161*a*c^2*d + 437*a*c*d^2 + 67*a*d^3)*cos(f*x + e) + (140*a*c^3 - 308*a*c^2*d + 244*a*c*d^2 - 76
*a*d^3 + 8*(a*c*d^2 + 13*a*d^3)*cos(f*x + e)^3 - 4*(7*a*c^2*d + 90*a*c*d^2 - 13*a*d^3)*cos(f*x + e)^2 - (35*a*
c^3 + 469*a*c^2*d + 193*a*c*d^2 + 143*a*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 + 2
8*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 + 2
7*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))**(3/2)/(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))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 20.21, size = 807, normalized size = 3.52 \begin {gather*} \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {32\,a\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (c+13\,d\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{105\,d^2\,f\,{\left (c+d\right )}^4}-\frac {16\,a\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (9\,c^3-5\,c^2\,d+9\,c\,d^2-d^3\right )}{3\,d^4\,f\,{\left (c+d\right )}^4}-\frac {16\,a\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^3\,9{}\mathrm {i}-c^2\,d\,5{}\mathrm {i}+c\,d^2\,9{}\mathrm {i}-d^3\,1{}\mathrm {i}\right )}{3\,d^4\,f\,{\left (c+d\right )}^4}-\frac {16\,a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (5\,c^3+65\,c^2\,d+c\,d^2+13\,d^3\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}-\frac {16\,a\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^3\,5{}\mathrm {i}+c^2\,d\,65{}\mathrm {i}+c\,d^2\,1{}\mathrm {i}+d^3\,13{}\mathrm {i}\right )}{15\,d^4\,f\,{\left (c+d\right )}^4}+\frac {32\,a\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (c\,1{}\mathrm {i}+d\,13{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{105\,d^2\,f\,{\left (c+d\right )}^4}+\frac {32\,a\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\left (c\,1{}\mathrm {i}+d\,13{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,f\,{\left (c+d\right )}^4}+\frac {32\,a\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (c+13\,d\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,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))^(3/2)/(c + d*sin(e + f*x))^(9/2),x)

[Out]

((c + d*sin(e + f*x))^(1/2)*((32*a*exp(e*8i + f*x*8i)*(c + 13*d)*(a + a*sin(e + f*x))^(1/2))/(105*d^2*f*(c + d
)^4) - (16*a*exp(e*4i + f*x*4i)*(a + a*sin(e + f*x))^(1/2)*(9*c*d^2 - 5*c^2*d + 9*c^3 - d^3))/(3*d^4*f*(c + d)
^4) - (16*a*exp(e*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2*9i - c^2*d*5i + c^3*9i - d^3*1i))/(3*d^4*f*(c
+ d)^4) - (16*a*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2 + 65*c^2*d + 5*c^3 + 13*d^3))/(15*d^4*f*
(c + d)^4) - (16*a*exp(e*3i + f*x*3i)*(a + a*sin(e + f*x))^(1/2)*(c*d^2*1i + c^2*d*65i + c^3*5i + d^3*13i))/(1
5*d^4*f*(c + d)^4) + (32*a*exp(e*1i + f*x*1i)*(c*1i + d*13i)*(a + a*sin(e + f*x))^(1/2))/(105*d^2*f*(c + d)^4)
+ (32*a*c*exp(e*7i + f*x*7i)*(c*1i + d*13i)*(a + a*sin(e + f*x))^(1/2))/(15*d^3*f*(c + d)^4) + (32*a*c*exp(e*
2i + f*x*2i)*(c + 13*d)*(a + a*sin(e + f*x))^(1/2))/(15*d^3*f*(c + d)^4)))/(exp(e*9i + f*x*9i) + ((c*1i + d*1i
)^4*1i)/(c + d)^4 - (4*exp(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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