3.4.74 \(\int \frac {(a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\) [374]

Optimal. Leaf size=184 \[ -\frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}} \]

[Out]

-a^2*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f/(c-c*sin(f*x+e))^(1/2)-1/3*a*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/f/(c-c
*sin(f*x+e))^(1/2)-8*a^4*cos(f*x+e)*ln(1-sin(f*x+e))/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-4*a^3*cos
(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f/(c-c*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2819, 2816, 2746, 31} \begin {gather*} -\frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {4 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 2816

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

Rule 2819

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

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx &=-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}+(2 a) \int \frac {(a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}+\left (4 a^2\right ) \int \frac {(a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}+\left (8 a^3\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}+\frac {\left (8 a^4 c \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}-\frac {\left (8 a^4 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{f \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 0.65, size = 150, normalized size = 0.82 \begin {gather*} -\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \sqrt {a (1+\sin (e+f x))} \left (-12 \cos (2 (e+f x))+192 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+87 \sin (e+f x)-\sin (3 (e+f x))\right )}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \sqrt {c-c \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-12*Cos[2*(e
+ f*x)] + 192*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 87*Sin[e + f*x] - Sin[3*(e + f*x)]))/(f*(Cos[(e + f*
x)/2] + Sin[(e + f*x)/2])^7*Sqrt[c - c*Sin[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(365\) vs. \(2(166)=332\).
time = 16.55, size = 366, normalized size = 1.99

method	result	size
default	\(-\frac {\left (\cos ^{4}\left (f x +e \right )-\left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-6 \left (\cos ^{3}\left (f x +e \right )\right )-5 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+48 \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right ) \cos \left (f x +e \right )+48 \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right ) \sin \left (f x +e \right )-17 \left (\cos ^{2}\left (f x +e \right )\right )+22 \cos \left (f x +e \right ) \sin \left (f x +e \right )-24 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \cos \left (f x +e \right )-24 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \sin \left (f x +e \right )-48 \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+6 \cos \left (f x +e \right )-16 \sin \left (f x +e \right )+24 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+16\right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}}}{3 f \left (\cos ^{4}\left (f x +e \right )+\left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \left (\cos ^{3}\left (f x +e \right )\right )-4 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-8 \left (\cos ^{2}\left (f x +e \right )\right )-4 \cos \left (f x +e \right ) \sin \left (f x +e \right )-4 \cos \left (f x +e \right )+8 \sin \left (f x +e \right )+8\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\)	\(366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/f*(cos(f*x+e)^4-cos(f*x+e)^3*sin(f*x+e)-6*cos(f*x+e)^3-5*sin(f*x+e)*cos(f*x+e)^2+48*ln(-(-1+cos(f*x+e)+si
n(f*x+e))/sin(f*x+e))*cos(f*x+e)+48*ln(-(-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))*sin(f*x+e)-17*cos(f*x+e)^2+22*c
os(f*x+e)*sin(f*x+e)-24*ln(2/(cos(f*x+e)+1))*cos(f*x+e)-24*ln(2/(cos(f*x+e)+1))*sin(f*x+e)-48*ln(-(-1+cos(f*x+
e)+sin(f*x+e))/sin(f*x+e))+6*cos(f*x+e)-16*sin(f*x+e)+24*ln(2/(cos(f*x+e)+1))+16)*(a*(1+sin(f*x+e)))^(7/2)/(co
s(f*x+e)^4+cos(f*x+e)^3*sin(f*x+e)+3*cos(f*x+e)^3-4*sin(f*x+e)*cos(f*x+e)^2-8*cos(f*x+e)^2-4*cos(f*x+e)*sin(f*
x+e)-4*cos(f*x+e)+8*sin(f*x+e)+8)/(-c*(sin(f*x+e)-1))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)/sqrt(-c*sin(f*x + e) + c), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*s
qrt(-c*sin(f*x + e) + c)/(c*sin(f*x + e) - c), x)

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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))**(7/2)/(c-c*sin(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [A]
time = 0.54, size = 148, normalized size = 0.80 \begin {gather*} \frac {4 \, a^{\frac {7}{2}} \sqrt {c} {\left (\frac {6 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {2 \, c^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 3 \, c^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 6 \, c^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{3 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*a^(7/2)*sqrt(c)*(6*log(-cos(-1/4*pi + 1/2*f*x + 1/2*e)^2 + 1)/(c*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + (2
*c^2*cos(-1/4*pi + 1/2*f*x + 1/2*e)^6 + 3*c^2*cos(-1/4*pi + 1/2*f*x + 1/2*e)^4 + 6*c^2*cos(-1/4*pi + 1/2*f*x +
1/2*e)^2)/(c^3*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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