\(\int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx\) [283]

Optimal result

Integrand size = 25, antiderivative size = 315 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx=\frac {c^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} b d^{5/2}}-\frac {c^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} b d^{5/2}}-\frac {c^{5/2} \log \left (\sqrt {c}-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{2 \sqrt {2} b d^{5/2}}+\frac {c^{5/2} \log \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{2 \sqrt {2} b d^{5/2}}+\frac {2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}} \]

[Out]

2/3*c*(c*sin(b*x+a))^(3/2)/b/d/(d*cos(b*x+a))^(3/2)+1/2*c^(5/2)*arctan(1-2^(1/2)*d^(1/2)*(c*sin(b*x+a))^(1/2)/
c^(1/2)/(d*cos(b*x+a))^(1/2))/b/d^(5/2)*2^(1/2)-1/2*c^(5/2)*arctan(1+2^(1/2)*d^(1/2)*(c*sin(b*x+a))^(1/2)/c^(1
/2)/(d*cos(b*x+a))^(1/2))/b/d^(5/2)*2^(1/2)-1/4*c^(5/2)*ln(c^(1/2)-2^(1/2)*d^(1/2)*(c*sin(b*x+a))^(1/2)/(d*cos
(b*x+a))^(1/2)+c^(1/2)*tan(b*x+a))/b/d^(5/2)*2^(1/2)+1/4*c^(5/2)*ln(c^(1/2)+2^(1/2)*d^(1/2)*(c*sin(b*x+a))^(1/
2)/(d*cos(b*x+a))^(1/2)+c^(1/2)*tan(b*x+a))/b/d^(5/2)*2^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 315, 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, 2654, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx=\frac {c^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} b d^{5/2}}-\frac {c^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}+1\right )}{\sqrt {2} b d^{5/2}}-\frac {c^{5/2} \log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)+\sqrt {c}\right )}{2 \sqrt {2} b d^{5/2}}+\frac {c^{5/2} \log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)+\sqrt {c}\right )}{2 \sqrt {2} b d^{5/2}}+\frac {2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}} \]

[In]

Int[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(5/2),x]

[Out]

(c^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(Sqrt[2]*b*d^(5/2)
) - (c^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(Sqrt[2]*b*d^(
5/2)) - (c^(5/2)*Log[Sqrt[c] - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + Sqrt[c]*Tan[a + b
*x]])/(2*Sqrt[2]*b*d^(5/2)) + (c^(5/2)*Log[Sqrt[c] + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x
]] + Sqrt[c]*Tan[a + b*x]])/(2*Sqrt[2]*b*d^(5/2)) + (2*c*(c*Sin[a + b*x])^(3/2))/(3*b*d*(d*Cos[a + b*x])^(3/2)
)

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 2654

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), 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*Sin[e + f*x])^(1/k)/(b*Cos
[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 (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}-\frac {c^2 \int \frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}} \, dx}{d^2} \\ & = \frac {2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}-\frac {\left (2 c^3\right ) \text {Subst}\left (\int \frac {x^2}{c^2+d^2 x^4} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{b d} \\ & = \frac {2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}+\frac {c^3 \text {Subst}\left (\int \frac {c-d x^2}{c^2+d^2 x^4} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{b d^2}-\frac {c^3 \text {Subst}\left (\int \frac {c+d x^2}{c^2+d^2 x^4} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{b d^2} \\ & = \frac {2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}-\frac {c^3 \text {Subst}\left (\int \frac {1}{\frac {c}{d}-\frac {\sqrt {2} \sqrt {c} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{2 b d^3}-\frac {c^3 \text {Subst}\left (\int \frac {1}{\frac {c}{d}+\frac {\sqrt {2} \sqrt {c} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{2 b d^3}-\frac {c^{5/2} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c}}{\sqrt {d}}+2 x}{-\frac {c}{d}-\frac {\sqrt {2} \sqrt {c} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{2 \sqrt {2} b d^{5/2}}-\frac {c^{5/2} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c}}{\sqrt {d}}-2 x}{-\frac {c}{d}+\frac {\sqrt {2} \sqrt {c} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}\right )}{2 \sqrt {2} b d^{5/2}} \\ & = -\frac {c^{5/2} \log \left (\sqrt {c}-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{2 \sqrt {2} b d^{5/2}}+\frac {c^{5/2} \log \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{2 \sqrt {2} b d^{5/2}}+\frac {2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}-\frac {c^{5/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} b d^{5/2}}+\frac {c^{5/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} b d^{5/2}} \\ & = \frac {c^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} b d^{5/2}}-\frac {c^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {c} \sqrt {d \cos (a+b x)}}\right )}{\sqrt {2} b d^{5/2}}-\frac {c^{5/2} \log \left (\sqrt {c}-\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{2 \sqrt {2} b d^{5/2}}+\frac {c^{5/2} \log \left (\sqrt {c}+\frac {\sqrt {2} \sqrt {d} \sqrt {c \sin (a+b x)}}{\sqrt {d \cos (a+b x)}}+\sqrt {c} \tan (a+b x)\right )}{2 \sqrt {2} b d^{5/2}}+\frac {2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.21 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx=\frac {2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},\frac {7}{4},\frac {11}{4},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{7/2}}{7 b c d (d \cos (a+b x))^{3/2}} \]

[In]

Integrate[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(5/2),x]

[Out]

(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[7/4, 7/4, 11/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(7/2))/(7*b*c*d*(
d*Cos[a + b*x])^(3/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(491\) vs. \(2(238)=476\).

Time = 0.26 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.56

method	result	size
default	\(\frac {\sqrt {2}\, \left (6 \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right ) \cos \left (b x +a \right )+6 \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right ) \cos \left (b x +a \right )-3 \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \ln \left (2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \cot \left (b x +a \right )+2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \csc \left (b x +a \right )+2-2 \cot \left (b x +a \right )\right ) \cos \left (b x +a \right )+3 \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \ln \left (-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \cot \left (b x +a \right )-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (1+\cos \left (b x +a \right )\right )^{2}}}\, \csc \left (b x +a \right )+2-2 \cot \left (b x +a \right )\right ) \cos \left (b x +a \right )-4 \sqrt {2}\, \cos \left (b x +a \right )+4 \sqrt {2}\right ) c^{2} \sqrt {c \sin \left (b x +a \right )}\, \left (1+\cos \left (b x +a \right )\right ) \sec \left (b x +a \right ) \csc \left (b x +a \right )}{12 b \sqrt {d \cos \left (b x +a \right )}\, d^{2}}\)	\(492\)

[In]

int((c*sin(b*x+a))^(5/2)/(d*cos(b*x+a))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/12/b*2^(1/2)*(6*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*arctan((sin(b*x+a)*2^(1/2)*(-sin(b*x+a)*cos(
b*x+a)/(1+cos(b*x+a))^2)^(1/2)+cos(b*x+a)-1)/(cos(b*x+a)-1))*cos(b*x+a)+6*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a
))^2)^(1/2)*arctan((sin(b*x+a)*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)-cos(b*x+a)+1)/(cos(b*x+
a)-1))*cos(b*x+a)-3*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*ln(2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+co
s(b*x+a))^2)^(1/2)*cot(b*x+a)+2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*csc(b*x+a)+2-2*cot(b*x
+a))*cos(b*x+a)+3*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*ln(-2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos
(b*x+a))^2)^(1/2)*cot(b*x+a)-2*2^(1/2)*(-sin(b*x+a)*cos(b*x+a)/(1+cos(b*x+a))^2)^(1/2)*csc(b*x+a)+2-2*cot(b*x+
a))*cos(b*x+a)-4*2^(1/2)*cos(b*x+a)+4*2^(1/2))*c^2*(c*sin(b*x+a))^(1/2)*(1+cos(b*x+a))/(d*cos(b*x+a))^(1/2)/d^
2*sec(b*x+a)*csc(b*x+a)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.50 (sec) , antiderivative size = 1164, normalized size of antiderivative = 3.70 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((c*sin(b*x+a))^(5/2)/(d*cos(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

1/24*(3*b*d^3*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a)^2*log(-1/2*c^8*cos(b*x + a)*sin(b*x + a) + 1/2*(b^3*d^7*(-
c^10/(b^4*d^10))^(3/4)*cos(b*x + a) - b*c^5*d^2*(-c^10/(b^4*d^10))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sq
rt(c*sin(b*x + a)) + 1/4*(2*b^2*c^3*d^5*cos(b*x + a)^2 - b^2*c^3*d^5)*sqrt(-c^10/(b^4*d^10))) - 3*b*d^3*(-c^10
/(b^4*d^10))^(1/4)*cos(b*x + a)^2*log(-1/2*c^8*cos(b*x + a)*sin(b*x + a) - 1/2*(b^3*d^7*(-c^10/(b^4*d^10))^(3/
4)*cos(b*x + a) - b*c^5*d^2*(-c^10/(b^4*d^10))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)) +
1/4*(2*b^2*c^3*d^5*cos(b*x + a)^2 - b^2*c^3*d^5)*sqrt(-c^10/(b^4*d^10))) - 3*I*b*d^3*(-c^10/(b^4*d^10))^(1/4)
*cos(b*x + a)^2*log(-1/2*c^8*cos(b*x + a)*sin(b*x + a) + 1/2*(I*b^3*d^7*(-c^10/(b^4*d^10))^(3/4)*cos(b*x + a)
+ I*b*c^5*d^2*(-c^10/(b^4*d^10))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)) - 1/4*(2*b^2*c^
3*d^5*cos(b*x + a)^2 - b^2*c^3*d^5)*sqrt(-c^10/(b^4*d^10))) + 3*I*b*d^3*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a)^
2*log(-1/2*c^8*cos(b*x + a)*sin(b*x + a) + 1/2*(-I*b^3*d^7*(-c^10/(b^4*d^10))^(3/4)*cos(b*x + a) - I*b*c^5*d^2
*(-c^10/(b^4*d^10))^(1/4)*sin(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a)) - 1/4*(2*b^2*c^3*d^5*cos(b*x
+ a)^2 - b^2*c^3*d^5)*sqrt(-c^10/(b^4*d^10))) - 3*b*d^3*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a)^2*log(c^8 + 2*(
b^3*d^7*(-c^10/(b^4*d^10))^(3/4)*sin(b*x + a) - b*c^5*d^2*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a))*sqrt(d*cos(b*
x + a))*sqrt(c*sin(b*x + a))) + 3*b*d^3*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a)^2*log(c^8 - 2*(b^3*d^7*(-c^10/(b
^4*d^10))^(3/4)*sin(b*x + a) - b*c^5*d^2*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*si
n(b*x + a))) - 3*I*b*d^3*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a)^2*log(c^8 - 2*(I*b^3*d^7*(-c^10/(b^4*d^10))^(3/
4)*sin(b*x + a) + I*b*c^5*d^2*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))
) + 3*I*b*d^3*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a)^2*log(c^8 - 2*(-I*b^3*d^7*(-c^10/(b^4*d^10))^(3/4)*sin(b*x
+ a) - I*b*c^5*d^2*(-c^10/(b^4*d^10))^(1/4)*cos(b*x + a))*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))) + 16*sqr
t(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*c^2*sin(b*x + a))/(b*d^3*cos(b*x + a)^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((c*sin(b*x+a))**(5/2)/(d*cos(b*x+a))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((c*sin(b*x+a))^(5/2)/(d*cos(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(5/2)/(d*cos(b*x + a))^(5/2), x)

Giac [F]

\[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((c*sin(b*x+a))^(5/2)/(d*cos(b*x+a))^(5/2),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^(5/2)/(d*cos(b*x + a))^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^{5/2}}{{\left (d\,\cos \left (a+b\,x\right )\right )}^{5/2}} \,d x \]

[In]

int((c*sin(a + b*x))^(5/2)/(d*cos(a + b*x))^(5/2),x)

[Out]

int((c*sin(a + b*x))^(5/2)/(d*cos(a + b*x))^(5/2), x)