∫ (c+d sin(e+fx))3∕2 ________________________________________

3.6.89 \(\int \frac {(c+d \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx\) [589]

Optimal. Leaf size=188 \[ -\frac {(3 c-d) \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {\sqrt {2} (c-d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}} \]

[Out]

-(c-d)^(3/2)*arctanh(1/2*cos(f*x+e)*a^(1/2)*(c-d)^(1/2)*2^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2))

*2^(1/2)/f/a^(1/2)-(3*c-d)*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^

(1/2)/f/a^(1/2)-d*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.40, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2857, 3061, 2861, 214, 2854, 211} \begin {gather*} -\frac {\sqrt {d} (3 c-d) \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 )}{\sqrt {a} f}-\frac {d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}-\frac {\sqrt {2} (c-d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(((3*c - d)*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])

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

+ f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f) - (d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[

e + f*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 214

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

x] && NegQ[a/b]

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]

Rule 2857

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp

[-2*d*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n - 1)/(f*(2*n - 1)*Sqrt[a + b*Sin[e + f*x]])), x] - Dist[1/(b*(2*n

- 1)), Int[((c + d*Sin[e + f*x])^(n - 2)/Sqrt[a + b*Sin[e + f*x]])*Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n -

1)) + d*(a*d - b*c*(4*n - 3))*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[n, 1] && IntegerQ[2*n]

Rule 2861

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D

ist[-2*(a/f), Subst[Int[1/(2*b^2 - (a*c - 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]

Rule 3061

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin

[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A*b - a*B)/b, Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*

x]]), x], x] + Dist[B/b, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e

, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 15.45, size = 1639, normalized size = 8.72 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-d \cos \left (\frac {1}{2} (e+f x)\right )+d \sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a (1+\sin (e+f x))}}+\frac {\left (\sqrt {2} (c-d)^{3/2} \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\sqrt {2} (c-d)^{3/2} \log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )+\frac {1}{2} i \sqrt {d} (-3 c+d) \left (\log \left (\frac {2 i \left (i c+d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(c+i d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^{3/2} (-3 c+d) \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )-\log \left (-\frac {2 \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^{3/2} (-3 c+d) \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\frac {c^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)}}-\frac {c d}{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)}}+\frac {d^2}{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)}}+\frac {3 c d \sin (e+f x)}{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)}}-\frac {d^2 \sin (e+f x)}{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)}}\right )}{f \sqrt {a (1+\sin (e+f x))} \left (\frac {(c-d)^{3/2} \sec ^2\left (\frac {1}{2} (e+f x)\right )}{\sqrt {2} \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {\sqrt {2} (c-d)^{3/2} \left (\frac {1}{2} (-c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} d \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {c+d \sin (e+f x)}}+\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}\right )}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}+\frac {1}{2} i \sqrt {d} (-3 c+d) \left (-\frac {i d^{3/2} (-3 c+d) \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (\frac {2 i \left (\frac {1}{2} (c+i d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {(1+i) d^{3/2} \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {2} \sqrt {c+d \sin (e+f x)}}+\frac {(1+i) \sqrt {d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {2}}\right )}{d^{3/2} (-3 c+d) \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {i \sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (i c+d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(c+i d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^{3/2} (-3 c+d) \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}\right )}{2 \left (i c+d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(c+i d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {d^{3/2} (-3 c+d) \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (-\frac {2 \left (\frac {1}{2} (i c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {(1+i) d^{3/2} \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {2} \sqrt {c+d \sin (e+f x)}}+\frac {(1+i) \sqrt {d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {2}}\right )}{d^{3/2} (-3 c+d) \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^{3/2} (-3 c+d) \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}\right )}{2 \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-(d*Cos[(e + f*x)/2]) + d*Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])/

(f*Sqrt[a*(1 + Sin[e + f*x])]) + ((Sqrt[2]*(c - d)^(3/2)*Log[1 + Tan[(e + f*x)/2]] - Sqrt[2]*(c - d)^(3/2)*Log

[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]] + (

I/2)*Sqrt[d]*(-3*c + d)*(Log[((2*I)*(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c +

d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-3*c + d)*(I + Tan[(e + f*x)/2]))] - Log[(-2*(c + I*d

+ (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]

))/(d^(3/2)*(-3*c + d)*(-I + Tan[(e + f*x)/2]))]))*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c^2/((Cos[(e + f*x)/

2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (c*d)/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*S

in[e + f*x]]) + d^2/(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) + (3*c*d*Sin[e + f*x])/

(2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]) - (d^2*Sin[e + f*x])/(2*(Cos[(e + f*x)/2] +

Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]])))/(f*Sqrt[a*(1 + Sin[e + f*x])]*(((c - d)^(3/2)*Sec[(e + f*x)/2]^

2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[2]*(c - d)^(3/2)*(((-c + d)*Sec[(e + f*x)/2]^2)/2 + (Sqrt[c - d]*d

*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/Sqrt[c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^

(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]))/(c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*

Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]) + (I/2)*Sqrt[d]*(-3*c + d)*(((-1/2*I)*d^(3/2)*(-3*c + d)*(I + Tan[(

e + f*x)/2])*(((2*I)*(((c + I*d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])

^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqr

t[c + d*Sin[e + f*x]])/Sqrt[2]))/(d^(3/2)*(-3*c + d)*(I + Tan[(e + f*x)/2])) - (I*Sec[(e + f*x)/2]^2*(I*c + d

+ (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2])

)/(d^(3/2)*(-3*c + d)*(I + Tan[(e + f*x)/2])^2)))/(I*c + d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(

-1)]*Sqrt[c + d*Sin[e + f*x]] + (c + I*d)*Tan[(e + f*x)/2]) + (d^(3/2)*(-3*c + d)*(-I + Tan[(e + f*x)/2])*((-2

*(((I*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqr

t[c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]

])/Sqrt[2]))/(d^(3/2)*(-3*c + d)*(-I + Tan[(e + f*x)/2])) + (Sec[(e + f*x)/2]^2*(c + I*d + (1 + I)*Sqrt[2]*Sqr

t[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-3*c + d)

*(-I + Tan[(e + f*x)/2])^2)))/(2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*S

in[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2])))))

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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{\sqrt {a +a \sin \left (f x +e \right )}}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2),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((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (166) = 332\).
time = 0.73, size = 2655, normalized size = 14.12 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[-1/8*(4*sqrt(2)*(a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*c - a*d)*sin(f*x + e))*sqrt((c - d)/a)*log(-(2*sqr

t(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) + (c

- 3*d)*cos(f*x + e)^2 + (3*c - d)*cos(f*x + e) - ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d

)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + (3*a*c - a*d + (3*a*c - a*d)*cos(f*

x + e) + (3*a*c - a*d)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)

^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d

- 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 + 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 -

c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^

2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*c

os(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*si

n(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(

f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*

c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x

+ e) + sin(f*x + e) + 1)) + 8*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x +

e) + c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f), -1/4*(2*sqrt(2)*(a*c - a*d + (a*c - a*d)*cos(f*x + e) +

(a*c - a*d)*sin(f*x + e))*sqrt((c - d)/a)*log(-(2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*s

qrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) + (c - 3*d)*cos(f*x + e)^2 + (3*c - d)*cos(f*x + e) - ((c - 3

*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - co

s(f*x + e) - 2)) - (3*a*c - a*d + (3*a*c - a*d)*cos(f*x + e) + (3*a*c - a*d)*sin(f*x + e))*sqrt(d/a)*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*si

n(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 +

2*d^3)*cos(f*x + e))) + 4*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e)

+ c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f), -1/8*(8*sqrt(2)*(a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*

c - a*d)*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt

(-(c - d)/a)/((c - d)*cos(f*x + e))) + (3*a*c - a*d + (3*a*c - a*d)*cos(f*x + e) + (3*a*c - a*d)*sin(f*x + e))

*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c

*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(

f*x + e)^2 + 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 -

2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*c

os(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^

2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 -

28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d

^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(

c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(d*co

s(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*cos(f*x + e) + a*f*si

n(f*x + e) + a*f), -1/4*(4*sqrt(2)*(a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*c - a*d)*sin(f*x + e))*sqrt(-(c

- d)/a)*arctan(-sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x +

e))) - (3*a*c - a*d + (3*a*c - a*d)*cos(f*x + e) + (3*a*c - a*d)*sin(f*x + e))*sqrt(d/a)*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(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(

f*x + e))) + 4*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*c

os(f*x + e) + a*f*sin(f*x + e) + a*f)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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