\(\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx\) [267]
Optimal result
Integrand size = 25, antiderivative size = 131 \[
\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx=\frac {c d \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{6 b}-\frac {c (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b d}+\frac {c^2 d^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{12 b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}
\]
[Out]
-1/3*c*(d*cos(b*x+a))^(5/2)*(c*sin(b*x+a))^(1/2)/b/d+1/6*c*d*(d*cos(b*x+a))^(1/2)*(c*sin(b*x+a))^(1/2)/b-1/12*
c^2*d^2*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(a+1/4*Pi+b*x),2^(1/2))*sin(2*b*x+2*a)^(1/2
)/b/(d*cos(b*x+a))^(1/2)/(c*sin(b*x+a))^(1/2)
Rubi [A] (verified)
Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2648, 2649, 2653, 2720}
\[
\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx=\frac {c^2 d^2 \sqrt {\sin (2 a+2 b x)} \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{12 b \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}-\frac {c \sqrt {c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b d}+\frac {c d \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{6 b}
\]
[In]
Int[(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2),x]
[Out]
(c*d*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])/(6*b) - (c*(d*Cos[a + b*x])^(5/2)*Sqrt[c*Sin[a + b*x]])/(3*b*d
) + (c^2*d^2*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(12*b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*
x]])
Rule 2648
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-a)*(b*Cos[e
+ f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Dist[a^2*((m - 1)/(m + n)), Int[(b*Cos[e + f*x
])^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[
2*m, 2*n]
Rule 2649
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(b*Sin[e +
f*x])^(n + 1)*((a*Cos[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Dist[a^2*((m - 1)/(m + n)), Int[(b*Sin[e + f*x])^
n*(a*Cos[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m
, 2*n]
Rule 2653
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, x]
Rule 2720
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]
Rubi steps \begin{align*}
\text {integral}& = -\frac {c (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b d}+\frac {1}{6} c^2 \int \frac {(d \cos (a+b x))^{3/2}}{\sqrt {c \sin (a+b x)}} \, dx \\ & = \frac {c d \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{6 b}-\frac {c (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b d}+\frac {1}{12} \left (c^2 d^2\right ) \int \frac {1}{\sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \, dx \\ & = \frac {c d \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{6 b}-\frac {c (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b d}+\frac {\left (c^2 d^2 \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{12 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \\ & = \frac {c d \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}}{6 b}-\frac {c (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)}}{3 b d}+\frac {c^2 d^2 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)}}{12 b \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}} \\
\end{align*}
Mathematica [C] (verified)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.54
\[
\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx=\frac {2 c d \sqrt {d \cos (a+b x)} \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {5}{4},\frac {9}{4},\sin ^2(a+b x)\right ) \sqrt {c \sin (a+b x)} \tan ^2(a+b x)}{5 b}
\]
[In]
Integrate[(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2),x]
[Out]
(2*c*d*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, 5/4, 9/4, Sin[a + b*x]^2]*Sqrt[c*Si
n[a + b*x]]*Tan[a + b*x]^2)/(5*b)
Maple [C] (warning: unable to verify)
Result contains complex when optimal does not.
Time = 1.47 (sec) , antiderivative size = 1744, normalized size of antiderivative =
13.31
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\(1744\) |
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[In]
int((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
[Out]
-1/48/b*2^(1/2)*(6*I*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^
(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(b*x+a)-6*I*(-cot(b*x+a)+csc(b*x+a
)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)
^(1/2),1/2-1/2*I,1/2*2^(1/2))+8*2^(1/2)*cos(b*x+a)^3*sin(b*x+a)-6*I*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+
a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2-1/2*I,1/2
*2^(1/2))*cos(b*x+a)+6*I*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+
a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-6*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2
)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/
2-1/2*I,1/2*2^(1/2))*cos(b*x+a)+8*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)
-csc(b*x+a))^(1/2)*EllipticF((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2*2^(1/2))*cos(b*x+a)-6*(-cot(b*x+a)+csc(b*x+a
)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)
^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(b*x+a)-6*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(c
ot(b*x+a)-csc(b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+8*(-cot(b*x+a)+
csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((-cot(b*x+a)+csc(b
*x+a)+1)^(1/2),1/2*2^(1/2))-6*(-cot(b*x+a)+csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc(b*x+a)+1)^(1/2)*(cot(b*x+a)-csc
(b*x+a))^(1/2)*EllipticPi((-cot(b*x+a)+csc(b*x+a)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-4*2^(1/2)*cos(b*x+a)*sin(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+co
s(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+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)-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)+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))+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))+3*(-sin(b*x+a)*co
s(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))-3*(-sin(b*x+a)*cos(b*x+a)/(1+c
os(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)))*(c*sin(b*x+a))^(1/2)*(d*cos(b*x+a))^(1/2)
*c*d*sec(b*x+a)*csc(b*x+a)
Fricas [F]
\[
\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \,d x }
\]
[In]
integrate((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^(3/2),x, algorithm="fricas")
[Out]
integral(sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*c*d*cos(b*x + a)*sin(b*x + a), x)
Sympy [F(-1)]
Timed out. \[
\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx=\text {Timed out}
\]
[In]
integrate((d*cos(b*x+a))**(3/2)*(c*sin(b*x+a))**(3/2),x)
[Out]
Timed out
Maxima [F]
\[
\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \,d x }
\]
[In]
integrate((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^(3/2),x, algorithm="maxima")
[Out]
integrate((d*cos(b*x + a))^(3/2)*(c*sin(b*x + a))^(3/2), x)
Giac [F]
\[
\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {3}{2}} \,d x }
\]
[In]
integrate((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^(3/2),x, algorithm="giac")
[Out]
integrate((d*cos(b*x + a))^(3/2)*(c*sin(b*x + a))^(3/2), x)
Mupad [F(-1)]
Timed out. \[
\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx=\int {\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2} \,d x
\]
[In]
int((d*cos(a + b*x))^(3/2)*(c*sin(a + b*x))^(3/2),x)
[Out]
int((d*cos(a + b*x))^(3/2)*(c*sin(a + b*x))^(3/2), x)