3.550 \(\int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=192 \[ \frac{d^2 (3 c-7 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{6 a^2 f}-\frac{(c+11 d) (c-d)^2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{2 \sqrt{2} a^{3/2} f}+\frac{d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt{a \sin (e+f x)+a}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}} \]

[Out]

-((c - d)^2*(c + 11*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*
f) + (d*(3*c^2 - 24*c*d + 13*d^2)*Cos[e + f*x])/(3*a*f*Sqrt[a + a*Sin[e + f*x]]) + ((3*c - 7*d)*d^2*Cos[e + f*
x]*Sqrt[a + a*Sin[e + f*x]])/(6*a^2*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*f*(a + a*Sin[e + f*x
])^(3/2))

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Rubi [A]  time = 0.462758, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2765, 2968, 3023, 2751, 2649, 206} \[ \frac{d^2 (3 c-7 d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{6 a^2 f}-\frac{(c+11 d) (c-d)^2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{2 \sqrt{2} a^{3/2} f}+\frac{d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt{a \sin (e+f x)+a}}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c - d)^2*(c + 11*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*
f) + (d*(3*c^2 - 24*c*d + 13*d^2)*Cos[e + f*x])/(3*a*f*Sqrt[a + a*Sin[e + f*x]]) + ((3*c - 7*d)*d^2*Cos[e + f*
x]*Sqrt[a + a*Sin[e + f*x]])/(6*a^2*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*f*(a + a*Sin[e + f*x
])^(3/2))

Rule 2765

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

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2751

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

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^{3/2}} \, dx &=-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac{\int \frac{(c+d \sin (e+f x)) \left (-\frac{1}{2} a \left (c^2+7 c d-4 d^2\right )+\frac{1}{2} a (3 c-7 d) d \sin (e+f x)\right )}{\sqrt{a+a \sin (e+f x)}} \, dx}{2 a^2}\\ &=-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{2} a c \left (c^2+7 c d-4 d^2\right )+\left (\frac{1}{2} a c (3 c-7 d) d-\frac{1}{2} a d \left (c^2+7 c d-4 d^2\right )\right ) \sin (e+f x)+\frac{1}{2} a (3 c-7 d) d^2 \sin ^2(e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx}{2 a^2}\\ &=\frac{(3 c-7 d) d^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{6 a^2 f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{4} a^2 \left (3 c^3+21 c^2 d-15 c d^2+7 d^3\right )+\frac{1}{2} a^2 d \left (3 c^2-24 c d+13 d^2\right ) \sin (e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx}{3 a^3}\\ &=\frac{d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt{a+a \sin (e+f x)}}+\frac{(3 c-7 d) d^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{6 a^2 f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}+\frac{\left ((c-d)^2 (c+11 d)\right ) \int \frac{1}{\sqrt{a+a \sin (e+f x)}} \, dx}{4 a}\\ &=\frac{d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt{a+a \sin (e+f x)}}+\frac{(3 c-7 d) d^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{6 a^2 f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac{\left ((c-d)^2 (c+11 d)\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{2 a f}\\ &=-\frac{(c-d)^2 (c+11 d) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a+a \sin (e+f x)}}\right )}{2 \sqrt{2} a^{3/2} f}+\frac{d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt{a+a \sin (e+f x)}}+\frac{(3 c-7 d) d^2 \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{6 a^2 f}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.54195, size = 328, normalized size = 1.71 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (-18 d^2 (2 c-d) \cos \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2+18 d^2 (2 c-d) \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2+6 (c-d)^3 \sin \left (\frac{1}{2} (e+f x)\right )-3 (c-d)^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+(3+3 i) (-1)^{3/4} (c+11 d) (c-d)^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (e+f x)\right )-1\right )\right )-2 d^3 \cos \left (\frac{3}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2-2 d^3 \sin \left (\frac{3}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2\right )}{6 f (a (\sin (e+f x)+1))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(6*(c - d)^3*Sin[(e + f*x)/2] - 3*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2]) + (3 + 3*I)*(-1)^(3/4)*(c - d)^2*(c + 11*d)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(e + f*x)/4])]
*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 18*(2*c - d)*d^2*Cos[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)
/2])^2 - 2*d^3*Cos[(3*(e + f*x))/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 18*(2*c - d)*d^2*Sin[(e + f*x)/2
]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*d^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sin[(3*(e + f*x))/2]
))/(6*f*(a*(1 + Sin[e + f*x]))^(3/2))

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Maple [B]  time = 0.769, size = 490, normalized size = 2.6 \begin{align*} -{\frac{1}{12\,f\cos \left ( fx+e \right ) } \left ( \sin \left ( fx+e \right ) \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}{c}^{3}+27\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}{c}^{2}d-63\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}c{d}^{2}+33\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}{d}^{3}-8\, \left ( a-a\sin \left ( fx+e \right ) \right ) ^{3/2}{d}^{3}\sqrt{a}+72\,{a}^{3/2}c{d}^{2}\sqrt{a-a\sin \left ( fx+e \right ) }-24\,{d}^{3}{a}^{3/2}\sqrt{a-a\sin \left ( fx+e \right ) } \right ) +3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}{c}^{3}+27\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}{c}^{2}d-63\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}c{d}^{2}+33\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{2}{d}^{3}-8\, \left ( a-a\sin \left ( fx+e \right ) \right ) ^{3/2}{d}^{3}\sqrt{a}+6\,\sqrt{a-a\sin \left ( fx+e \right ) }{a}^{3/2}{c}^{3}-18\,\sqrt{a-a\sin \left ( fx+e \right ) }{a}^{3/2}{c}^{2}d+90\,{a}^{3/2}c{d}^{2}\sqrt{a-a\sin \left ( fx+e \right ) }-30\,{d}^{3}{a}^{3/2}\sqrt{a-a\sin \left ( fx+e \right ) } \right ) \sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(sin(f*x+e)*(3*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^3+27*2^(1/2)*arctanh(1/
2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^2*d-63*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1
/2))*a^2*c*d^2+33*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*d^3-8*(a-a*sin(f*x+e))^(3/2)
*d^3*a^(1/2)+72*a^(3/2)*c*d^2*(a-a*sin(f*x+e))^(1/2)-24*d^3*a^(3/2)*(a-a*sin(f*x+e))^(1/2))+3*2^(1/2)*arctanh(
1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*c^3+27*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1
/2))*a^2*c^2*d-63*2^(1/2)*arctanh(1/2*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*c*d^2+33*2^(1/2)*arctanh(1/2
*(a-a*sin(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*a^2*d^3-8*(a-a*sin(f*x+e))^(3/2)*d^3*a^(1/2)+6*(a-a*sin(f*x+e))^(1/2)
*a^(3/2)*c^3-18*(a-a*sin(f*x+e))^(1/2)*a^(3/2)*c^2*d+90*a^(3/2)*c*d^2*(a-a*sin(f*x+e))^(1/2)-30*d^3*a^(3/2)*(a
-a*sin(f*x+e))^(1/2))*(-a*(-1+sin(f*x+e)))^(1/2)/a^(7/2)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*sin(f*x + e) + c)^3/(a*sin(f*x + e) + a)^(3/2), x)

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Fricas [B]  time = 1.76277, size = 1211, normalized size = 6.31 \begin{align*} -\frac{3 \, \sqrt{2}{\left (2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} -{\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right )^{2} +{\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right ) +{\left (2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} +{\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt{a} \log \left (-\frac{a \cos \left (f x + e\right )^{2} - 2 \, \sqrt{2} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{a}{\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) -{\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} -{\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \,{\left (4 \, d^{3} \cos \left (f x + e\right )^{3} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} - 4 \,{\left (9 \, c d^{2} - 4 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \,{\left (c^{3} - 3 \, c^{2} d + 15 \, c d^{2} - 5 \, d^{3}\right )} \cos \left (f x + e\right ) -{\left (4 \, d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} + 12 \,{\left (3 \, c d^{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}}{24 \,{\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f -{\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(3*sqrt(2)*(2*c^3 + 18*c^2*d - 42*c*d^2 + 22*d^3 - (c^3 + 9*c^2*d - 21*c*d^2 + 11*d^3)*cos(f*x + e)^2 +
(c^3 + 9*c^2*d - 21*c*d^2 + 11*d^3)*cos(f*x + e) + (2*c^3 + 18*c^2*d - 42*c*d^2 + 22*d^3 + (c^3 + 9*c^2*d - 21
*c*d^2 + 11*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) +
a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(
cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(4*d^3*cos(f*x + e)^3 - 3*c^3 + 9*c^
2*d - 9*c*d^2 + 3*d^3 - 4*(9*c*d^2 - 4*d^3)*cos(f*x + e)^2 - 3*(c^3 - 3*c^2*d + 15*c*d^2 - 5*d^3)*cos(f*x + e)
- (4*d^3*cos(f*x + e)^2 - 3*c^3 + 9*c^2*d - 9*c*d^2 + 3*d^3 + 12*(3*c*d^2 - d^3)*cos(f*x + e))*sin(f*x + e))*
sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f
)*sin(f*x + e))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))**3/(a+a*sin(f*x+e))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage2