\(\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} (A+C \sec ^2(c+d x)) \, dx\) [165]

Optimal result

Integrand size = 35, antiderivative size = 174 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^2 (63 A+47 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (63 A+47 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 (63 A+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d} \]

[Out]

2/315*(63*A+22*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2/9*C*sec(d*x+c)^2*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/d+2
/21*C*(a+a*sec(d*x+c))^(5/2)*tan(d*x+c)/a/d+8/315*a^2*(63*A+47*C)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+2/315*a*
(63*A+47*C)*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4174, 4095, 4086, 3878, 3877} \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^2 (63 A+47 C) \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {2 (63 A+22 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{315 d}+\frac {2 a (63 A+47 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{21 a d} \]

[In]

Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(8*a^2*(63*A + 47*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(63*A + 47*C)*Sqrt[a + a*Sec[c + d*
x]]*Tan[c + d*x])/(315*d) + (2*(63*A + 22*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d
*x]^2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(9*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(21*a*d)

Rule 3877

Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*b*(Cot[e + f*x]/(
f*Sqrt[a + b*Csc[e + f*x]])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rule 3878

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-b)*Cot[e + f*x]*(
(a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Dist[a*((2*m - 1)/m), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1),
x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && IntegerQ[2*m]

Rule 4086

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))
, x_Symbol] :> Simp[(-B)*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*B*m + A*b*(m + 1))/(b
*(m + 1)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, A, B, e, f, m}, x] && NeQ[A*b - a*B
, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] &&  !LtQ[m, -2^(-1)]

Rule 4095

Int[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_
)), x_Symbol] :> Simp[(-B)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(b*(m + 2)),
Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*B*(m + 1) + (A*b*(m + 2) - a*B)*Csc[e + f*x], x], x], x] /; Fr
eeQ[{a, b, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] &&  !LtQ[m, -1]

Rule 4174

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*(m + n + 1
))), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n*Simp[A*b*(m + n + 1) + b*C*n +
a*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(
-1)] &&  !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (9 A+4 C)+\frac {3}{2} a C \sec (c+d x)\right ) \, dx}{9 a} \\ & = \frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}+\frac {4 \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {15 a^2 C}{4}+\frac {1}{4} a^2 (63 A+22 C) \sec (c+d x)\right ) \, dx}{63 a^2} \\ & = \frac {2 (63 A+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}+\frac {1}{105} (63 A+47 C) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx \\ & = \frac {2 a (63 A+47 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 (63 A+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}+\frac {1}{315} (4 a (63 A+47 C)) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx \\ & = \frac {8 a^2 (63 A+47 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a (63 A+47 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 (63 A+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac {2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.52 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a^2 \left (378 A+272 C+(189 A+136 C) \sec (c+d x)+3 (21 A+34 C) \sec ^2(c+d x)+85 C \sec ^3(c+d x)+35 C \sec ^4(c+d x)\right ) \tan (c+d x)}{315 d \sqrt {a (1+\sec (c+d x))}} \]

[In]

Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]

[Out]

(2*a^2*(378*A + 272*C + (189*A + 136*C)*Sec[c + d*x] + 3*(21*A + 34*C)*Sec[c + d*x]^2 + 85*C*Sec[c + d*x]^3 +
35*C*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {2 a \left (378 A \cos \left (d x +c \right )^{4}+272 C \cos \left (d x +c \right )^{4}+189 A \cos \left (d x +c \right )^{3}+136 C \cos \left (d x +c \right )^{3}+63 A \cos \left (d x +c \right )^{2}+102 C \cos \left (d x +c \right )^{2}+85 C \cos \left (d x +c \right )+35 C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{315 d \left (\cos \left (d x +c \right )+1\right )}\)	\(122\)
parts	\(\frac {2 A a \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (6 \sin \left (d x +c \right )+3 \tan \left (d x +c \right )+\sec \left (d x +c \right ) \tan \left (d x +c \right )\right )}{5 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 C a \left (272 \cos \left (d x +c \right )^{4}+136 \cos \left (d x +c \right )^{3}+102 \cos \left (d x +c \right )^{2}+85 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{315 d \left (\cos \left (d x +c \right )+1\right )}\)	\(144\)

[In]

int(sec(d*x+c)^2*(a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

2/315*a/d*(378*A*cos(d*x+c)^4+272*C*cos(d*x+c)^4+189*A*cos(d*x+c)^3+136*C*cos(d*x+c)^3+63*A*cos(d*x+c)^2+102*C
*cos(d*x+c)^2+85*C*cos(d*x+c)+35*C)*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)*tan(d*x+c)*sec(d*x+c)^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (2 \, {\left (189 \, A + 136 \, C\right )} a \cos \left (d x + c\right )^{4} + {\left (189 \, A + 136 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 34 \, C\right )} a \cos \left (d x + c\right )^{2} + 85 \, C a \cos \left (d x + c\right ) + 35 \, C a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \]

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

2/315*(2*(189*A + 136*C)*a*cos(d*x + c)^4 + (189*A + 136*C)*a*cos(d*x + c)^3 + 3*(21*A + 34*C)*a*cos(d*x + c)^
2 + 85*C*a*cos(d*x + c) + 35*C*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/(d*cos(d*x + c)^5 + d*c
os(d*x + c)^4)

Sympy [F]

\[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]

[In]

integrate(sec(d*x+c)**2*(a+a*sec(d*x+c))**(3/2)*(A+C*sec(d*x+c)**2),x)

[Out]

Integral((a*(sec(c + d*x) + 1))**(3/2)*(A + C*sec(c + d*x)**2)*sec(c + d*x)**2, x)

Maxima [F(-1)]

Timed out. \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(sec(d*x+c)^2*(a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 24.39 (sec) , antiderivative size = 621, normalized size of antiderivative = 3.57 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a\,\left (5\,A+4\,C\right )\,4{}\mathrm {i}}{9\,d}-\frac {a\,\left (7\,A+12\,C\right )\,4{}\mathrm {i}}{9\,d}+\frac {A\,a\,8{}\mathrm {i}}{9\,d}\right )-\frac {a\,\left (5\,A+4\,C\right )\,4{}\mathrm {i}}{9\,d}+\frac {a\,\left (7\,A+12\,C\right )\,4{}\mathrm {i}}{9\,d}-\frac {A\,a\,8{}\mathrm {i}}{9\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a\,\left (3\,A+4\,C\right )\,4{}\mathrm {i}}{5\,d}-\frac {A\,a\,4{}\mathrm {i}}{5\,d}+\frac {C\,a\,16{}\mathrm {i}}{105\,d}\right )-\frac {A\,a\,12{}\mathrm {i}}{5\,d}+\frac {a\,\left (A+12\,C\right )\,4{}\mathrm {i}}{5\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a\,\left (3\,A+8\,C\right )\,4{}\mathrm {i}}{7\,d}-\frac {a\,\left (A+C\right )\,16{}\mathrm {i}}{7\,d}+\frac {A\,a\,4{}\mathrm {i}}{7\,d}+\frac {C\,a\,32{}\mathrm {i}}{63\,d}\right )+\frac {A\,a\,12{}\mathrm {i}}{7\,d}-\frac {a\,\left (A+3\,C\right )\,16{}\mathrm {i}}{7\,d}+\frac {a\,\left (A-8\,C\right )\,4{}\mathrm {i}}{7\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {a\,\left (21\,A+34\,C\right )\,8{}\mathrm {i}}{315\,d}-\frac {A\,a\,4{}\mathrm {i}}{3\,d}\right )-\frac {A\,a\,4{}\mathrm {i}}{d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}-\frac {a\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (189\,A+136\,C\right )\,4{}\mathrm {i}}{315\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )} \]

[In]

int(((A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^(3/2))/cos(c + d*x)^2,x)

[Out]

((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((a*(5*A + 4*C)*4i)/(9*d) -
(a*(7*A + 12*C)*4i)/(9*d) + (A*a*8i)/(9*d)) - (a*(5*A + 4*C)*4i)/(9*d) + (a*(7*A + 12*C)*4i)/(9*d) - (A*a*8i)
/(9*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^4) + ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i +
d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((a*(3*A + 4*C)*4i)/(5*d) - (A*a*4i)/(5*d) + (C*a*16i)/(105*d)) - (A*a*1
2i)/(5*d) + (a*(A + 12*C)*4i)/(5*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^2) + ((a + a/(exp(- c
*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((a*(3*A + 8*C)*4i)/(7*d) - (a*(A + C)*16i)
/(7*d) + (A*a*4i)/(7*d) + (C*a*32i)/(63*d)) + (A*a*12i)/(7*d) - (a*(A + 3*C)*16i)/(7*d) + (a*(A - 8*C)*4i)/(7*
d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^3) - ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*
1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((a*(21*A + 34*C)*8i)/(315*d) - (A*a*4i)/(3*d)) - (A*a*4i)/d))/((exp(c*1i +
d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)) - (a*exp(c*1i + d*x*1i)*(a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x
*1i)/2))^(1/2)*(189*A + 136*C)*4i)/(315*d*(exp(c*1i + d*x*1i) + 1))