3.6.72 \(\int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2} \, dx\) [572]

Optimal. Leaf size=228 \[ \frac {a^{3/2} (c-11 d) (c+d)^2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{3/2} f}+\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}} \]

[Out]

1/8*a^(3/2)*(c-11*d)*(c+d)^2*arctan(cos(f*x+e)*a^(1/2)*d^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))/
d^(3/2)/f+1/12*a^2*(c-11*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d/f/(a+a*sin(f*x+e))^(1/2)-1/3*a^2*cos(f*x+e)*(c
+d*sin(f*x+e))^(5/2)/d/f/(a+a*sin(f*x+e))^(1/2)+1/8*a^2*(c-11*d)*(c+d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d/f/(
a+a*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.29, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2842, 21, 2849, 2854, 211} \begin {gather*} \frac {a^{3/2} (c-11 d) (c+d)^2 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{3/2} f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a \sin (e+f x)+a}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a \sin (e+f x)+a}}+\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a \sin (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(3/2)*(c - 11*d)*(c + d)^2*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]])])/(8*d^(3/2)*f) + (a^2*(c - 11*d)*(c + d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8*d*f*Sqrt[a + a*S
in[e + f*x]]) + (a^2*(c - 11*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*d*f*Sqrt[a + a*Sin[e + f*x]]) - (
a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(3*d*f*Sqrt[a + a*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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2842

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/(
d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d
*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, 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 2849

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

Rule 2854

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*(b/f), Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(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]

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2} \, dx &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}+\frac {\int \frac {\left (-\frac {1}{2} a^2 (c-11 d)-\frac {1}{2} a^2 (c-11 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{3 d}\\ &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-11 d)) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx}{6 d}\\ &=\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-11 d) (c+d)) \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx}{8 d}\\ &=\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}-\frac {\left (a (c-11 d) (c+d)^2\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{16 d}\\ &=\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c-11 d) (c+d)^2\right ) \text {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d f}\\ &=\frac {a^{3/2} (c-11 d) (c+d)^2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{3/2} f}+\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 0.65, size = 281, normalized size = 1.23 \begin {gather*} \frac {(a (1+\sin (e+f x)))^{3/2} \left (\frac {(c-11 d) (c+d)^2 \left (-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )+\log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right )\right )}{d^{3/2}}-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} \left (3 c^2+52 c d+37 d^2-4 d^2 \cos (2 (e+f x))+2 d (7 c+11 d) \sin (e+f x)\right )}{3 d}\right )}{16 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(1 + Sin[e + f*x]))^(3/2)*(((c - 11*d)*(c + d)^2*(-2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/Sq
rt[c + d*Sin[e + f*x]]] - ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + Log[
Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(3/2) - (2*(Cos[(e + f*x)/2] - Sin[(
e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(3*c^2 + 52*c*d + 37*d^2 - 4*d^2*Cos[2*(e + f*x)] + 2*d*(7*c + 11*d)*Sin
[e + f*x]))/(3*d)))/(16*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}\, dx\]

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))^(3/2),x)

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (208) = 416\).
time = 0.76, size = 1391, normalized size = 6.10 \begin {gather*} \left [-\frac {3 \, {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3} + {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right )} \cos \left (f x + e\right ) + {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {a}{d}} \log \left (\frac {128 \, a d^{4} \cos \left (f x + e\right )^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left (2 \, a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{4} - 32 \, {\left (5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, d^{4} \cos \left (f x + e\right )^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left (c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - {\left (c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right )} \cos \left (f x + e\right ) + {\left (16 \, d^{4} \cos \left (f x + e\right )^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left (3 \, c d^{3} - 5 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\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} \sqrt {-\frac {a}{d}} + {\left (a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right )} \cos \left (f x + e\right ) + {\left (128 \, a d^{4} \cos \left (f x + e\right )^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left (a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right ) - 8 \, {\left (8 \, a d^{2} \cos \left (f x + e\right )^{3} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} - 14 \, {\left (a c d + a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (3 \, a c^{2} + 52 \, a c d + 41 \, a d^{2}\right )} \cos \left (f x + e\right ) - {\left (8 \, a d^{2} \cos \left (f x + e\right )^{2} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} + 2 \, {\left (7 \, a c d + 11 \, a 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}}{192 \, {\left (d f \cos \left (f x + e\right ) + d f \sin \left (f x + e\right ) + d f\right )}}, -\frac {3 \, {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3} + {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right )} \cos \left (f x + e\right ) + {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {a}{d}} \arctan \left (\frac {{\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left (c d - d^{2}\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} \sqrt {\frac {a}{d}}}{4 \, {\left (2 \, a d^{2} \cos \left (f x + e\right )^{3} - {\left (3 \, a c d - a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a c^{2} - a c d + 2 \, a d^{2}\right )} \cos \left (f x + e\right )\right )}}\right ) - 4 \, {\left (8 \, a d^{2} \cos \left (f x + e\right )^{3} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} - 14 \, {\left (a c d + a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (3 \, a c^{2} + 52 \, a c d + 41 \, a d^{2}\right )} \cos \left (f x + e\right ) - {\left (8 \, a d^{2} \cos \left (f x + e\right )^{2} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} + 2 \, {\left (7 \, a c d + 11 \, a 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}}{96 \, {\left (d f \cos \left (f x + e\right ) + d f \sin \left (f x + e\right ) + d f\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))^(3/2),x, algorithm="fricas")

[Out]

[-1/192*(3*(a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3 + (a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3)*cos(f*x +
e) + (a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^
4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 1
4*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16
*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2
- 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x +
e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3
+ 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 -
28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c
^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 +
5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f
*x + e) + sin(f*x + e) + 1)) - 8*(8*a*d^2*cos(f*x + e)^3 - 3*a*c^2 - 38*a*c*d - 19*a*d^2 - 14*(a*c*d + a*d^2)*
cos(f*x + e)^2 - (3*a*c^2 + 52*a*c*d + 41*a*d^2)*cos(f*x + e) - (8*a*d^2*cos(f*x + e)^2 - 3*a*c^2 - 38*a*c*d -
19*a*d^2 + 2*(7*a*c*d + 11*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) +
c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f), -1/96*(3*(a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3 + (a*c^3
- 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3)*cos(f*x + e) + (a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3)*sin(f*x + e))*
sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x
+ e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x
+ e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) - 4*(8*a*d^2*cos(f*x + e)^3 - 3*a*c^2 - 38*a*c*d - 19*a*d^2 -
14*(a*c*d + a*d^2)*cos(f*x + e)^2 - (3*a*c^2 + 52*a*c*d + 41*a*d^2)*cos(f*x + e) - (8*a*d^2*cos(f*x + e)^2 - 3
*a*c^2 - 38*a*c*d - 19*a*d^2 + 2*(7*a*c*d + 11*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqr
t(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \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))**(3/2),x)

[Out]

Integral((a*(sin(e + f*x) + 1))**(3/2)*(c + d*sin(e + f*x))**(3/2), x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \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))^(3/2),x)

[Out]

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

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