\(\int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx\) [272]

Optimal result

Integrand size = 25, antiderivative size = 313 \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=-\frac {c^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} b d^{3/2}}+\frac {c^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} b d^{3/2}}+\frac {c^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {c^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}} \]

[Out]

1/2*c^(3/2)*arctan(-1+2^(1/2)*c^(1/2)*(d*cos(b*x+a))^(1/2)/d^(1/2)/(c*sin(b*x+a))^(1/2))/b/d^(3/2)*2^(1/2)+1/2
*c^(3/2)*arctan(1+2^(1/2)*c^(1/2)*(d*cos(b*x+a))^(1/2)/d^(1/2)/(c*sin(b*x+a))^(1/2))/b/d^(3/2)*2^(1/2)+1/4*c^(
3/2)*ln(d^(1/2)+cot(b*x+a)*d^(1/2)-2^(1/2)*c^(1/2)*(d*cos(b*x+a))^(1/2)/(c*sin(b*x+a))^(1/2))/b/d^(3/2)*2^(1/2
)-1/4*c^(3/2)*ln(d^(1/2)+cot(b*x+a)*d^(1/2)+2^(1/2)*c^(1/2)*(d*cos(b*x+a))^(1/2)/(c*sin(b*x+a))^(1/2))/b/d^(3/
2)*2^(1/2)+2*c*(c*sin(b*x+a))^(1/2)/b/d/(d*cos(b*x+a))^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2646, 2655, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=-\frac {c^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} b d^{3/2}}+\frac {c^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}+1\right )}{\sqrt {2} b d^{3/2}}+\frac {c^{3/2} \log \left (-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+\sqrt {d} \cot (a+b x)+\sqrt {d}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {c^{3/2} \log \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}+\sqrt {d} \cot (a+b x)+\sqrt {d}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}} \]

[In]

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

[Out]

-((c^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])])/(Sqrt[2]*b*d^(3/
2))) + (c^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])])/(Sqrt[2]*b*
d^(3/2)) + (c^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[a + b*x] - (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/Sqrt[c*Sin[a +
b*x]]])/(2*Sqrt[2]*b*d^(3/2)) - (c^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[a + b*x] + (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a +
b*x]])/Sqrt[c*Sin[a + b*x]]])/(2*Sqrt[2]*b*d^(3/2)) + (2*c*Sqrt[c*Sin[a + b*x]])/(b*d*Sqrt[d*Cos[a + b*x]])

Rule 210

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

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2646

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

Rule 2655

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{k = Denomina
tor[m]}, Dist[(-k)*a*(b/f), Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*
Sin[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}-\frac {c^2 \int \frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}} \, dx}{d^2} \\ & = \frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}+\frac {\left (2 c^3\right ) \text {Subst}\left (\int \frac {x^2}{d^2+c^2 x^4} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{b d} \\ & = \frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}-\frac {c^2 \text {Subst}\left (\int \frac {d-c x^2}{d^2+c^2 x^4} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{b d}+\frac {c^2 \text {Subst}\left (\int \frac {d+c x^2}{d^2+c^2 x^4} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{b d} \\ & = \frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}+\frac {c^{3/2} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt {c}}+2 x}{-\frac {d}{c}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt {c}}-x^2} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {c^{3/2} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {d}}{\sqrt {c}}-2 x}{-\frac {d}{c}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt {c}}-x^2} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {c \text {Subst}\left (\int \frac {1}{\frac {d}{c}-\frac {\sqrt {2} \sqrt {d} x}{\sqrt {c}}+x^2} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 b d}+\frac {c \text {Subst}\left (\int \frac {1}{\frac {d}{c}+\frac {\sqrt {2} \sqrt {d} x}{\sqrt {c}}+x^2} \, dx,x,\frac {\sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 b d} \\ & = \frac {c^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {c^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}}+\frac {c^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} b d^{3/2}}-\frac {c^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} b d^{3/2}} \\ & = -\frac {c^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} b d^{3/2}}+\frac {c^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {d} \sqrt {c \sin (a+b x)}}\right )}{\sqrt {2} b d^{3/2}}+\frac {c^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)-\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {c^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (a+b x)+\frac {\sqrt {2} \sqrt {c} \sqrt {d \cos (a+b x)}}{\sqrt {c \sin (a+b x)}}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {2 c \sqrt {c \sin (a+b x)}}{b d \sqrt {d \cos (a+b x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.21 \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\frac {2 \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {5}{4},\frac {9}{4},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{5/2}}{5 b c d \sqrt {d \cos (a+b x)}} \]

[In]

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

[Out]

(2*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(5/2))/(5*b*c*d*Sq
rt[d*Cos[a + b*x]])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(686\) vs. \(2(237)=474\).

Time = 0.25 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.19

method	result	size
default	\(-\frac {\sqrt {2}\, \left (\frac {c \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )+1}\right )^{\frac {3}{2}} \left (\sin ^{2}\left (b x +a \right )\right ) \left (\ln \left (-\frac {-\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )+2 \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )-2+2 \cos \left (b x +a \right )+\sin \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right ) \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}+2 \arctan \left (\frac {\sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{1-\cos \left (b x +a \right )}\right ) \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}-\ln \left (\frac {\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )+2 \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )+2-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right ) \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}+2 \arctan \left (\frac {\sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )+1-\cos \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right ) \sqrt {\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}+8 \csc \left (b x +a \right )-8 \cot \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right )}{4 b \left (1-\cos \left (b x +a \right )\right )^{2} {\left (-\frac {d \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right )}{\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )+1}\right )}^{\frac {3}{2}}}\)	\(687\)

[In]

int((c*sin(b*x+a))^(3/2)/(d*cos(b*x+a))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/4/b*2^(1/2)*(c/((1-cos(b*x+a))^2*csc(b*x+a)^2+1)*(csc(b*x+a)-cot(b*x+a)))^(3/2)/(1-cos(b*x+a))^2*sin(b*x+a)
^2*(ln(-1/(1-cos(b*x+a))*(-(1-cos(b*x+a))^2*csc(b*x+a)+2*((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc
(b*x+a))^(1/2)*sin(b*x+a)-2+2*cos(b*x+a)+sin(b*x+a)))*((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*
x+a))^(1/2)+2*arctan(1/(1-cos(b*x+a))*(((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)*sin
(b*x+a)+cos(b*x+a)-1))*((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)-ln(1/(1-cos(b*x+a))
*((1-cos(b*x+a))^2*csc(b*x+a)+2*((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)*sin(b*x+a)
+2-2*cos(b*x+a)-sin(b*x+a)))*((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)+2*arctan(1/(1
-cos(b*x+a))*(((1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)*sin(b*x+a)+1-cos(b*x+a)))*((
1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)+8*csc(b*x+a)-8*cot(b*x+a))*((1-cos(b*x+a))^2
*csc(b*x+a)^2-1)/(-d*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)/((1-cos(b*x+a))^2*csc(b*x+a)^2+1))^(3/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 1092, normalized size of antiderivative = 3.49 \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*(-I*b*d^2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(2*b^2*c^2*d^3*sqrt(-c^6/(b^4*d^6))*cos(b*x + a)*sin(b*x
+ a) + 2*c^5*cos(b*x + a)^2 - c^5 - 2*(I*b^3*d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) - I*b*c^3*d*(-c^6/(b^4*d
^6))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) + I*b*d^2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x +
a)*log(2*b^2*c^2*d^3*sqrt(-c^6/(b^4*d^6))*cos(b*x + a)*sin(b*x + a) + 2*c^5*cos(b*x + a)^2 - c^5 - 2*(-I*b^3*
d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) + I*b*c^3*d*(-c^6/(b^4*d^6))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*
sqrt(c*sin(b*x + a))) - b*d^2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(-2*b^2*c^2*d^3*sqrt(-c^6/(b^4*d^6))*cos(
b*x + a)*sin(b*x + a) + 2*c^5*cos(b*x + a)^2 - c^5 + 2*(b^3*d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) + b*c^3*d*
(-c^6/(b^4*d^6))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) + b*d^2*(-c^6/(b^4*d^6))^(1/4)
*cos(b*x + a)*log(-2*b^2*c^2*d^3*sqrt(-c^6/(b^4*d^6))*cos(b*x + a)*sin(b*x + a) + 2*c^5*cos(b*x + a)^2 - c^5 -
2*(b^3*d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) + b*c^3*d*(-c^6/(b^4*d^6))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x
+ a))*sqrt(c*sin(b*x + a))) + b*d^2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(-c^5 + 2*(b^3*d^4*(-c^6/(b^4*d^6))
^(3/4)*cos(b*x + a) - b*c^3*d*(-c^6/(b^4*d^6))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)))
- b*d^2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(-c^5 - 2*(b^3*d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) - b*c^3*
d*(-c^6/(b^4*d^6))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) + I*b*d^2*(-c^6/(b^4*d^6))^(
1/4)*cos(b*x + a)*log(-c^5 - 2*(I*b^3*d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) + I*b*c^3*d*(-c^6/(b^4*d^6))^(1/
4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) - I*b*d^2*(-c^6/(b^4*d^6))^(1/4)*cos(b*x + a)*log(
-c^5 - 2*(-I*b^3*d^4*(-c^6/(b^4*d^6))^(3/4)*cos(b*x + a) - I*b*c^3*d*(-c^6/(b^4*d^6))^(1/4)*sin(b*x + a))*sqrt
(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) - 16*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*c)/(b*d^2*cos(b*x + a))

Sympy [F]

\[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {\left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}}}{\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((c*sin(b*x+a))**(3/2)/(d*cos(b*x+a))**(3/2),x)

[Out]

Integral((c*sin(a + b*x))**(3/2)/(d*cos(a + b*x))**(3/2), x)

Maxima [F]

\[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((c*sin(b*x + a))^(3/2)/(d*cos(b*x + a))^(3/2), x)

Giac [F]

\[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((c*sin(b*x + a))^(3/2)/(d*cos(b*x + a))^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}} \,d x \]

[In]

int((c*sin(a + b*x))^(3/2)/(d*cos(a + b*x))^(3/2),x)

[Out]

int((c*sin(a + b*x))^(3/2)/(d*cos(a + b*x))^(3/2), x)