3.6.70 \(\int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx\) [570]

Optimal. Leaf size=142 \[ -\frac {2 a \cos (e+f x)}{5 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a \cos (e+f x)}{15 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a \cos (e+f x)}{15 (c+d)^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]

[Out]

-2/5*a*cos(f*x+e)/(c+d)/f/(c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)-8/15*a*cos(f*x+e)/(c+d)^2/f/(c+d*sin(f
*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-16/15*a*cos(f*x+e)/(c+d)^3/f/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2851, 2850} \begin {gather*} -\frac {16 a \cos (e+f x)}{15 f (c+d)^3 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {8 a \cos (e+f x)}{15 f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac {2 a \cos (e+f x)}{5 f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(-2*a*Cos[e + f*x])/(5*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a*Cos[e + f*x])/(15
*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a*Cos[e + f*x])/(15*(c + d)^3*f*Sqrt[a
+ a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])

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 {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx &=-\frac {2 a \cos (e+f x)}{5 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac {4 \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 (c+d)}\\ &=-\frac {2 a \cos (e+f x)}{5 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a \cos (e+f x)}{15 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {8 \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 (c+d)^2}\\ &=-\frac {2 a \cos (e+f x)}{5 (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a \cos (e+f x)}{15 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a \cos (e+f x)}{15 (c+d)^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 128, normalized size = 0.90 \begin {gather*} -\frac {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 (15 c^2+10 c d+3 d^2+4 d (5 c+d) \sin (e+f x)+8 d^2 \sin ^2(e+f x)\right )}{15 (c+d)^3 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))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x])^(7/2),x]

[Out]

(-2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(15*c^2 + 10*c*d + 3*d^2 + 4*d*(5*c + d)*
Sin[e + f*x] + 8*d^2*Sin[e + f*x]^2))/(15*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c + d*Sin[e + f*x
])^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(429\) vs. \(2(124)=248\).
time = 0.24, size = 430, normalized size = 3.03

method	result	size
default	\(\frac {2 \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (-15 c^{5}-7 d^{5}-6 c^{2} d^{3}-11 c \,d^{4}-22 c^{3} d^{2}-35 c^{4} d +15 c^{5} \sin \left (f x +e \right )-23 \left (\cos ^{4}\left (f x +e \right )\right ) d^{5}+8 \left (\cos ^{6}\left (f x +e \right )\right ) d^{5}+22 \left (\cos ^{2}\left (f x +e \right )\right ) d^{5}+25 \left (\cos ^{2}\left (f x +e \right )\right ) c^{4} d -11 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) d^{5}-15 \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}+19 \left (\cos ^{2}\left (f x +e \right )\right ) c^{3} d^{2}-2 \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{4}+21 \left (\cos ^{4}\left (f x +e \right )\right ) c^{2} d^{3}+13 \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}+4 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) d^{5}+4 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{4}+7 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{3} d^{2}+3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d^{3}-15 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{4}+7 \sin \left (f x +e \right ) d^{5}+35 \sin \left (f x +e \right ) c^{4} d +22 \sin \left (f x +e \right ) c^{3} d^{2}+6 \sin \left (f x +e \right ) c^{2} d^{3}+11 \sin \left (f x +e \right ) c \,d^{4}\right )}{15 f \cos \left (f x +e \right ) \left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right )^{3} \left (c +d \right )^{3}}\)	\(430\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/15/f*(a*(1+sin(f*x+e)))^(1/2)*(c+d*sin(f*x+e))^(1/2)*(-15*c^5-7*d^5-6*c^2*d^3-11*c*d^4-22*c^3*d^2-35*c^4*d+1
5*c^5*sin(f*x+e)+4*sin(f*x+e)*cos(f*x+e)^4*c*d^4+7*sin(f*x+e)*cos(f*x+e)^2*c^3*d^2+3*sin(f*x+e)*cos(f*x+e)^2*c
^2*d^3-15*sin(f*x+e)*cos(f*x+e)^2*c*d^4+22*cos(f*x+e)^2*d^5+7*sin(f*x+e)*d^5+8*cos(f*x+e)^6*d^5-23*cos(f*x+e)^
4*d^5+21*cos(f*x+e)^4*c^2*d^3-2*cos(f*x+e)^4*c*d^4-11*sin(f*x+e)*cos(f*x+e)^2*d^5+25*cos(f*x+e)^2*c^4*d+19*cos
(f*x+e)^2*c^3*d^2-15*cos(f*x+e)^2*c^2*d^3+13*cos(f*x+e)^2*c*d^4+35*sin(f*x+e)*c^4*d+22*sin(f*x+e)*c^3*d^2+6*si
n(f*x+e)*c^2*d^3+11*sin(f*x+e)*c*d^4+4*sin(f*x+e)*cos(f*x+e)^4*d^5)/cos(f*x+e)/(cos(f*x+e)^2*d^2+c^2-d^2)^3/(c
+d)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (133) = 266\).
time = 0.61, size = 570, normalized size = 4.01 \begin {gather*} -\frac {2 \, {\left ({\left (15 \, c^{3} + 10 \, c^{2} d + 3 \, c d^{2}\right )} \sqrt {a} - \frac {{\left (15 \, c^{3} - 60 \, c^{2} d - 25 \, c d^{2} - 6 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (45 \, c^{3} - 40 \, c^{2} d + 93 \, c d^{2} + 10 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5 \, {\left (9 \, c^{3} - 22 \, c^{2} d + 13 \, c d^{2} - 12 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, {\left (9 \, c^{3} - 22 \, c^{2} d + 13 \, c d^{2} - 12 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {{\left (45 \, c^{3} - 40 \, c^{2} d + 93 \, c d^{2} + 10 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {{\left (15 \, c^{3} - 60 \, c^{2} d - 25 \, c d^{2} - 6 \, d^{3}\right )} \sqrt {a} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {{\left (15 \, c^{3} + 10 \, c^{2} d + 3 \, c d^{2}\right )} \sqrt {a} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{3}}{15 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3} + \frac {3 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, {\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {{\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\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 {7}{2}} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

-2/15*((15*c^3 + 10*c^2*d + 3*c*d^2)*sqrt(a) - (15*c^3 - 60*c^2*d - 25*c*d^2 - 6*d^3)*sqrt(a)*sin(f*x + e)/(co
s(f*x + e) + 1) + (45*c^3 - 40*c^2*d + 93*c*d^2 + 10*d^3)*sqrt(a)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 5*(9*c
^3 - 22*c^2*d + 13*c*d^2 - 12*d^3)*sqrt(a)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*(9*c^3 - 22*c^2*d + 13*c*d^
2 - 12*d^3)*sqrt(a)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - (45*c^3 - 40*c^2*d + 93*c*d^2 + 10*d^3)*sqrt(a)*sin(
f*x + e)^5/(cos(f*x + e) + 1)^5 + (15*c^3 - 60*c^2*d - 25*c*d^2 - 6*d^3)*sqrt(a)*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - (15*c^3 + 10*c^2*d + 3*c*d^2)*sqrt(a)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*(sin(f*x + e)^2/(cos(f*x +
e) + 1)^2 + 1)^3/((c^3 + 3*c^2*d + 3*c*d^2 + d^3 + 3*(c^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x + e)^2/(cos(f*x
+ e) + 1)^2 + 3*(c^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (c^3 + 3*c^2*d + 3*c*d^2
+ d^3)*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)^(7/2)*f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (133) = 266\).
time = 0.38, size = 572, normalized size = 4.03 \begin {gather*} \frac {2 \, {\left (8 \, d^{2} \cos \left (f x + e\right )^{3} - 4 \, {\left (5 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} - 15 \, c^{2} + 10 \, c d - 7 \, d^{2} - {\left (15 \, c^{2} + 10 \, c d + 11 \, d^{2}\right )} \cos \left (f x + e\right ) - {\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - 15 \, c^{2} + 10 \, c d - 7 \, d^{2} + 4 \, {\left (5 \, c d + d^{2}\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}}{15 \, {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{4} - 3 \, {\left (c^{4} d^{2} + 3 \, c^{3} d^{3} + 3 \, c^{2} d^{4} + c d^{5}\right )} f \cos \left (f x + e\right )^{3} - {\left (3 \, c^{5} d + 12 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 18 \, c^{2} d^{4} + 9 \, c d^{5} + 2 \, d^{6}\right )} f \cos \left (f x + e\right )^{2} + {\left (c^{6} + 3 \, c^{5} d + 6 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 9 \, c^{2} d^{4} + 3 \, c d^{5}\right )} f \cos \left (f x + e\right ) + {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f - {\left ({\left (c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (3 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 12 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{5} d + 9 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right )} f \cos \left (f x + e\right ) - {\left (c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\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))^(1/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

2/15*(8*d^2*cos(f*x + e)^3 - 4*(5*c*d - d^2)*cos(f*x + e)^2 - 15*c^2 + 10*c*d - 7*d^2 - (15*c^2 + 10*c*d + 11*
d^2)*cos(f*x + e) - (8*d^2*cos(f*x + e)^2 - 15*c^2 + 10*c*d - 7*d^2 + 4*(5*c*d + d^2)*cos(f*x + e))*sin(f*x +
e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^4
- 3*(c^4*d^2 + 3*c^3*d^3 + 3*c^2*d^4 + c*d^5)*f*cos(f*x + e)^3 - (3*c^5*d + 12*c^4*d^2 + 20*c^3*d^3 + 18*c^2*d
^4 + 9*c*d^5 + 2*d^6)*f*cos(f*x + e)^2 + (c^6 + 3*c^5*d + 6*c^4*d^2 + 10*c^3*d^3 + 9*c^2*d^4 + 3*c*d^5)*f*cos(
f*x + e) + (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f - ((c^3*d^3 + 3*c^2*d^4 +
3*c*d^5 + d^6)*f*cos(f*x + e)^3 + (3*c^4*d^2 + 10*c^3*d^3 + 12*c^2*d^4 + 6*c*d^5 + d^6)*f*cos(f*x + e)^2 - (3*
c^5*d + 9*c^4*d^2 + 10*c^3*d^3 + 6*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e) - (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*
c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f)*sin(f*x + e))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(1/2)/(c+d*sin(f*x+e))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4369 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1060 vs. \(2 (133) = 266\).
time = 1.27, size = 1060, normalized size = 7.46 \begin {gather*} \frac {4 \, \sqrt {2} {\left ({\left ({\left (5 \, {\left (\frac {3 \, {\left (c^{10} d^{6} - 2 \, c^{9} d^{7} - 3 \, c^{8} d^{8} + 8 \, c^{7} d^{9} + 2 \, c^{6} d^{10} - 12 \, c^{5} d^{11} + 2 \, c^{4} d^{12} + 8 \, c^{3} d^{13} - 3 \, c^{2} d^{14} - 2 \, c d^{15} + d^{16}\right )} \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2}}{c^{11} d^{6} - c^{10} d^{7} - 5 \, c^{9} d^{8} + 5 \, c^{8} d^{9} + 10 \, c^{7} d^{10} - 10 \, c^{6} d^{11} - 10 \, c^{5} d^{12} + 10 \, c^{4} d^{13} + 5 \, c^{3} d^{14} - 5 \, c^{2} d^{15} - c d^{16} + d^{17}} + \frac {4 \, {\left (3 \, c^{10} d^{6} - 14 \, c^{9} d^{7} + 15 \, c^{8} d^{8} + 24 \, c^{7} d^{9} - 58 \, c^{6} d^{10} + 12 \, c^{5} d^{11} + 54 \, c^{4} d^{12} - 40 \, c^{3} d^{13} - 9 \, c^{2} d^{14} + 18 \, c d^{15} - 5 \, d^{16}\right )}}{c^{11} d^{6} - c^{10} d^{7} - 5 \, c^{9} d^{8} + 5 \, c^{8} d^{9} + 10 \, c^{7} d^{10} - 10 \, c^{6} d^{11} - 10 \, c^{5} d^{12} + 10 \, c^{4} d^{13} + 5 \, c^{3} d^{14} - 5 \, c^{2} d^{15} - c d^{16} + d^{17}}\right )} \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + \frac {2 \, {\left (45 \, c^{10} d^{6} - 250 \, c^{9} d^{7} + 601 \, c^{8} d^{8} - 664 \, c^{7} d^{9} - 166 \, c^{6} d^{10} + 1444 \, c^{5} d^{11} - 1510 \, c^{4} d^{12} + 104 \, c^{3} d^{13} + 889 \, c^{2} d^{14} - 634 \, c d^{15} + 141 \, d^{16}\right )}}{c^{11} d^{6} - c^{10} d^{7} - 5 \, c^{9} d^{8} + 5 \, c^{8} d^{9} + 10 \, c^{7} d^{10} - 10 \, c^{6} d^{11} - 10 \, c^{5} d^{12} + 10 \, c^{4} d^{13} + 5 \, c^{3} d^{14} - 5 \, c^{2} d^{15} - c d^{16} + d^{17}}\right )} \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + \frac {20 \, {\left (3 \, c^{10} d^{6} - 14 \, c^{9} d^{7} + 15 \, c^{8} d^{8} + 24 \, c^{7} d^{9} - 58 \, c^{6} d^{10} + 12 \, c^{5} d^{11} + 54 \, c^{4} d^{12} - 40 \, c^{3} d^{13} - 9 \, c^{2} d^{14} + 18 \, c d^{15} - 5 \, d^{16}\right )}}{c^{11} d^{6} - c^{10} d^{7} - 5 \, c^{9} d^{8} + 5 \, c^{8} d^{9} + 10 \, c^{7} d^{10} - 10 \, c^{6} d^{11} - 10 \, c^{5} d^{12} + 10 \, c^{4} d^{13} + 5 \, c^{3} d^{14} - 5 \, c^{2} d^{15} - c d^{16} + d^{17}}\right )} \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + \frac {15 \, {\left (c^{10} d^{6} - 2 \, c^{9} d^{7} - 3 \, c^{8} d^{8} + 8 \, c^{7} d^{9} + 2 \, c^{6} d^{10} - 12 \, c^{5} d^{11} + 2 \, c^{4} d^{12} + 8 \, c^{3} d^{13} - 3 \, c^{2} d^{14} - 2 \, c d^{15} + d^{16}\right )}}{c^{11} d^{6} - c^{10} d^{7} - 5 \, c^{9} d^{8} + 5 \, c^{8} d^{9} + 10 \, c^{7} d^{10} - 10 \, c^{6} d^{11} - 10 \, c^{5} d^{12} + 10 \, c^{4} d^{13} + 5 \, c^{3} d^{14} - 5 \, c^{2} d^{15} - c d^{16} + d^{17}}\right )} \sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )}{15 \, {\left (c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + 2 \, c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} - 6 \, d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + c + d\right )}^{\frac {5}{2}} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

4/15*sqrt(2)*(((5*(3*(c^10*d^6 - 2*c^9*d^7 - 3*c^8*d^8 + 8*c^7*d^9 + 2*c^6*d^10 - 12*c^5*d^11 + 2*c^4*d^12 + 8
*c^3*d^13 - 3*c^2*d^14 - 2*c*d^15 + d^16)*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2/(c^11*d^6 - c^10*d^7 - 5*c^9*d^8 +
5*c^8*d^9 + 10*c^7*d^10 - 10*c^6*d^11 - 10*c^5*d^12 + 10*c^4*d^13 + 5*c^3*d^14 - 5*c^2*d^15 - c*d^16 + d^17) +
4*(3*c^10*d^6 - 14*c^9*d^7 + 15*c^8*d^8 + 24*c^7*d^9 - 58*c^6*d^10 + 12*c^5*d^11 + 54*c^4*d^12 - 40*c^3*d^13
- 9*c^2*d^14 + 18*c*d^15 - 5*d^16)/(c^11*d^6 - c^10*d^7 - 5*c^9*d^8 + 5*c^8*d^9 + 10*c^7*d^10 - 10*c^6*d^11 -
10*c^5*d^12 + 10*c^4*d^13 + 5*c^3*d^14 - 5*c^2*d^15 - c*d^16 + d^17))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 2*(45
*c^10*d^6 - 250*c^9*d^7 + 601*c^8*d^8 - 664*c^7*d^9 - 166*c^6*d^10 + 1444*c^5*d^11 - 1510*c^4*d^12 + 104*c^3*d
^13 + 889*c^2*d^14 - 634*c*d^15 + 141*d^16)/(c^11*d^6 - c^10*d^7 - 5*c^9*d^8 + 5*c^8*d^9 + 10*c^7*d^10 - 10*c^
6*d^11 - 10*c^5*d^12 + 10*c^4*d^13 + 5*c^3*d^14 - 5*c^2*d^15 - c*d^16 + d^17))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^
2 + 20*(3*c^10*d^6 - 14*c^9*d^7 + 15*c^8*d^8 + 24*c^7*d^9 - 58*c^6*d^10 + 12*c^5*d^11 + 54*c^4*d^12 - 40*c^3*d
^13 - 9*c^2*d^14 + 18*c*d^15 - 5*d^16)/(c^11*d^6 - c^10*d^7 - 5*c^9*d^8 + 5*c^8*d^9 + 10*c^7*d^10 - 10*c^6*d^1
1 - 10*c^5*d^12 + 10*c^4*d^13 + 5*c^3*d^14 - 5*c^2*d^15 - c*d^16 + d^17))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 1
5*(c^10*d^6 - 2*c^9*d^7 - 3*c^8*d^8 + 8*c^7*d^9 + 2*c^6*d^10 - 12*c^5*d^11 + 2*c^4*d^12 + 8*c^3*d^13 - 3*c^2*d
^14 - 2*c*d^15 + d^16)/(c^11*d^6 - c^10*d^7 - 5*c^9*d^8 + 5*c^8*d^9 + 10*c^7*d^10 - 10*c^6*d^11 - 10*c^5*d^12
+ 10*c^4*d^13 + 5*c^3*d^14 - 5*c^2*d^15 - c*d^16 + d^17))*sqrt(a)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8
*pi + 1/4*f*x + 1/4*e)/((c*tan(-1/8*pi + 1/4*f*x + 1/4*e)^4 + d*tan(-1/8*pi + 1/4*f*x + 1/4*e)^4 + 2*c*tan(-1/
8*pi + 1/4*f*x + 1/4*e)^2 - 6*d*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + c + d)^(5/2)*f)

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Mupad [B]
time = 16.35, size = 501, normalized size = 3.53 \begin {gather*} -\frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,32{}\mathrm {i}}{15\,d\,f\,{\left (c+d\right )}^3}-\frac {32\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d\,f\,{\left (c+d\right )}^3}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (240\,c^2+80\,d^2\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,f\,{\left (c+d\right )}^3}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (c^2\,240{}\mathrm {i}+d^2\,80{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{15\,d^3\,f\,{\left (c+d\right )}^3}+\frac {32\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,d^2\,f\,{\left (c+d\right )}^3}-\frac {c\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,32{}\mathrm {i}}{3\,d^2\,f\,{\left (c+d\right )}^3}\right )}{{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3}{{\left (c+d\right )}^3}-\frac {3\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (6\,c+d\right )}{d}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (c\,6{}\mathrm {i}+d\,1{}\mathrm {i}\right )}{d}-\frac {3\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (4\,c^2+2\,c\,d+d^2\right )}{d^2\,{\left (c+d\right )}^3}+\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^3\,\left (8\,c^3+12\,c^2\,d+12\,c\,d^2+3\,d^3\right )}{d^3\,{\left (c+d\right )}^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(7/2),x)

[Out]

-((c + d*sin(e + f*x))^(1/2)*((exp(e*1i + f*x*1i)*(a + a*sin(e + f*x))^(1/2)*32i)/(15*d*f*(c + d)^3) - (32*exp
(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2))/(15*d*f*(c + d)^3) + (exp(e*4i + f*x*4i)*(240*c^2 + 80*d^2)*(a + a
*sin(e + f*x))^(1/2))/(15*d^3*f*(c + d)^3) - (exp(e*3i + f*x*3i)*(c^2*240i + d^2*80i)*(a + a*sin(e + f*x))^(1/
2))/(15*d^3*f*(c + d)^3) + (32*c*exp(e*2i + f*x*2i)*(a + a*sin(e + f*x))^(1/2))/(3*d^2*f*(c + d)^3) - (c*exp(e
*5i + f*x*5i)*(a + a*sin(e + f*x))^(1/2)*32i)/(3*d^2*f*(c + d)^3)))/(exp(e*7i + f*x*7i) + (c*1i + d*1i)^3/(c +
d)^3 - (3*exp(e*5i + f*x*5i)*(2*c*d + 4*c^2 + d^2))/d^2 - (exp(e*1i + f*x*1i)*(6*c + d))/d + (exp(e*3i + f*x*
3i)*(12*c*d^2 + 12*c^2*d + 8*c^3 + 3*d^3))/d^3 + (exp(e*6i + f*x*6i)*(c*6i + d*1i))/d - (3*exp(e*2i + f*x*2i)*
(c*1i + d*1i)^3*(2*c*d + 4*c^2 + d^2))/(d^2*(c + d)^3) + (exp(e*4i + f*x*4i)*(c*1i + d*1i)^3*(12*c*d^2 + 12*c^
2*d + 8*c^3 + 3*d^3))/(d^3*(c + d)^3))

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