Integrand size = 26, antiderivative size = 252 \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=-\frac {99 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {99 i a^2}{224 d (a+i a \tan (c+d x))^{7/2}}-\frac {i a^4}{4 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}+\frac {99 i a}{320 d (a+i a \tan (c+d x))^{5/2}}+\frac {33 i}{128 d (a+i a \tan (c+d x))^{3/2}}+\frac {99 i}{256 a d \sqrt {a+i a \tan (c+d x)}}-\frac {11 i a^5}{16 d (a+i a \tan (c+d x))^{7/2} \left (a^3-i a^3 \tan (c+d x)\right )} \] Output:
-99/512*I*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/a^ (3/2)/d+99/224*I*a^2/d/(a+I*a*tan(d*x+c))^(7/2)-1/4*I*a^4/d/(a-I*a*tan(d*x +c))^2/(a+I*a*tan(d*x+c))^(7/2)+99/320*I*a/d/(a+I*a*tan(d*x+c))^(5/2)+33/1 28*I/d/(a+I*a*tan(d*x+c))^(3/2)+99/256*I/a/d/(a+I*a*tan(d*x+c))^(1/2)-11/1 6*I*a^5/d/(a+I*a*tan(d*x+c))^(7/2)/(a^3-I*a^3*tan(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.21 \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {i a^2 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},3,-\frac {5}{2},\frac {1}{2} (1+i \tan (c+d x))\right )}{28 d (a+i a \tan (c+d x))^{7/2}} \] Input:
Integrate[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
((I/28)*a^2*Hypergeometric2F1[-7/2, 3, -5/2, (1 + I*Tan[c + d*x])/2])/(d*( a + I*a*Tan[c + d*x])^(7/2))
Time = 0.35 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3968, 52, 52, 61, 61, 61, 61, 73, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (c+d x)^4 (a+i a \tan (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3968 |
| \(\displaystyle -\frac {i a^5 \int \frac {1}{(a-i a \tan (c+d x))^3 (i \tan (c+d x) a+a)^{9/2}}d(i a \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {i a^5 \left (\frac {11 \int \frac {1}{(a-i a \tan (c+d x))^2 (i \tan (c+d x) a+a)^{9/2}}d(i a \tan (c+d x))}{8 a}+\frac {1}{4 a (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {i a^5 \left (\frac {11 \left (\frac {9 \int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{9/2}}d(i a \tan (c+d x))}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}\right )}{8 a}+\frac {1}{4 a (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {i a^5 \left (\frac {11 \left (\frac {9 \left (\frac {\int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{7/2}}d(i a \tan (c+d x))}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}\right )}{8 a}+\frac {1}{4 a (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {i a^5 \left (\frac {11 \left (\frac {9 \left (\frac {\frac {\int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{5/2}}d(i a \tan (c+d x))}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}\right )}{8 a}+\frac {1}{4 a (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {i a^5 \left (\frac {11 \left (\frac {9 \left (\frac {\frac {\frac {\int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{3/2}}d(i a \tan (c+d x))}{2 a}-\frac {1}{3 a (a+i a \tan (c+d x))^{3/2}}}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}\right )}{8 a}+\frac {1}{4 a (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {i a^5 \left (\frac {11 \left (\frac {9 \left (\frac {\frac {\frac {\frac {\int \frac {1}{(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}d(i a \tan (c+d x))}{2 a}-\frac {1}{a \sqrt {a+i a \tan (c+d x)}}}{2 a}-\frac {1}{3 a (a+i a \tan (c+d x))^{3/2}}}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}\right )}{8 a}+\frac {1}{4 a (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {i a^5 \left (\frac {11 \left (\frac {9 \left (\frac {\frac {\frac {\frac {\int \frac {1}{a^2 \tan ^2(c+d x)+2 a}d\sqrt {i \tan (c+d x) a+a}}{a}-\frac {1}{a \sqrt {a+i a \tan (c+d x)}}}{2 a}-\frac {1}{3 a (a+i a \tan (c+d x))^{3/2}}}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}\right )}{8 a}+\frac {1}{4 a (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {i a^5 \left (\frac {11 \left (\frac {9 \left (\frac {\frac {\frac {\frac {i \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2}}\right )}{\sqrt {2} a^{3/2}}-\frac {1}{a \sqrt {a+i a \tan (c+d x)}}}{2 a}-\frac {1}{3 a (a+i a \tan (c+d x))^{3/2}}}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{7/2}}\right )}{8 a}+\frac {1}{4 a (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{7/2}}\right )}{d}\) |
Input:
Int[Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^(3/2),x]
Output:
((-I)*a^5*(1/(4*a*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(7/2)) + (11*(1/(2*a*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(7/2)) + (9*(-1 /7*1/(a*(a + I*a*Tan[c + d*x])^(7/2)) + (-1/5*1/(a*(a + I*a*Tan[c + d*x])^ (5/2)) + (-1/3*1/(a*(a + I*a*Tan[c + d*x])^(3/2)) + ((I*ArcTan[(Sqrt[a]*Ta n[c + d*x])/Sqrt[2]])/(Sqrt[2]*a^(3/2)) - 1/(a*Sqrt[a + I*a*Tan[c + d*x]]) )/(2*a))/(2*a))/(2*a)))/(4*a)))/(8*a)))/d
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(a^(m - 2)*b*f) Subst[Int[(a - x)^(m/2 - 1)*(a + x )^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (201 ) = 402\).
Time = 8.86 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.67
| method | result | size |
| default | \(\frac {i \left (6930 \cos \left (d x +c \right )+3465\right ) \sin \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )+i}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+\left (6930 \cos \left (d x +c \right )^{2}+3465 \cos \left (d x +c \right )-3465\right ) \operatorname {arctanh}\left (\frac {\cot \left (d x +c \right )-\csc \left (d x +c \right )+i}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+3465 i \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-3465 \sin \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (3520 \cos \left (d x +c \right )^{4}+3520 \cos \left (d x +c \right )^{3}+5544 \cos \left (d x +c \right )^{2}+5544 \cos \left (d x +c \right )-6930\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+i \cos \left (d x +c \right ) \left (-960 \cos \left (d x +c \right )^{5}-960 \cos \left (d x +c \right )^{4}-2376 \cos \left (d x +c \right )^{3}-2376 \cos \left (d x +c \right )^{2}+11550 \cos \left (d x +c \right )+4620\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{17920 d \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (i \cos \left (d x +c \right ) \sin \left (d x +c \right )+i \sin \left (d x +c \right )+\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )\right ) a}\) | \(421\) |
Input:
int(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/17920/d*(I*(6930*cos(d*x+c)+3465)*sin(d*x+c)*arctanh(1/2/(-cos(d*x+c)/(c os(d*x+c)+1))^(1/2)*(cot(d*x+c)-csc(d*x+c)+I))+(6930*cos(d*x+c)^2+3465*cos (d*x+c)-3465)*arctanh(1/2/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(cot(d*x+c)-c sc(d*x+c)+I))+3465*I*cos(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/ 2)-3465*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+sin(d*x+c) *cos(d*x+c)*(3520*cos(d*x+c)^4+3520*cos(d*x+c)^3+5544*cos(d*x+c)^2+5544*co s(d*x+c)-6930)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+I*cos(d*x+c)*(-960*cos(d *x+c)^5-960*cos(d*x+c)^4-2376*cos(d*x+c)^3-2376*cos(d*x+c)^2+11550*cos(d*x +c)+4620)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))/(a*(1+I*tan(d*x+c)))^(1/2)/( -cos(d*x+c)/(cos(d*x+c)+1))^(1/2)/(I*sin(d*x+c)*cos(d*x+c)+I*sin(d*x+c)+co s(d*x+c)^2+cos(d*x+c))/a
Time = 0.09 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.25 \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {{\left (-3465 i \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 3465 i \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-70 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 805 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 2833 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 4584 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1304 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 328 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 40 i\right )}\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{17920 \, a^{2} d} \] Input:
integrate(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="fricas")
Output:
1/17920*(-3465*I*sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e^(7*I*d*x + 7*I*c)*log (4*(sqrt(2)*sqrt(1/2)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d *x + 2*I*c) + 1))*sqrt(1/(a^3*d^2)) + a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + 3465*I*sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2))*e^(7*I*d*x + 7*I*c)*log(-4*(sq rt(2)*sqrt(1/2)*(a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt(a/(e^(2*I*d*x + 2 *I*c) + 1))*sqrt(1/(a^3*d^2)) - a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + sqr t(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-70*I*e^(12*I*d*x + 12*I*c) - 805* I*e^(10*I*d*x + 10*I*c) + 2833*I*e^(8*I*d*x + 8*I*c) + 4584*I*e^(6*I*d*x + 6*I*c) + 1304*I*e^(4*I*d*x + 4*I*c) + 328*I*e^(2*I*d*x + 2*I*c) + 40*I))* e^(-7*I*d*x - 7*I*c)/(a^2*d)
\[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cos(d*x+c)**4/(a+I*a*tan(d*x+c))**(3/2),x)
Output:
Integral(cos(c + d*x)**4/(I*a*(tan(c + d*x) - I))**(3/2), x)
Time = 0.13 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.82 \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {i \, {\left (\frac {3465 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{\sqrt {a}} + \frac {4 \, {\left (3465 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{5} - 11550 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a + 7392 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{2} + 2112 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{3} + 1408 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} + 1280 \, a^{5}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}}\right )}}{35840 \, a d} \] Input:
integrate(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="maxima")
Output:
1/35840*I*(3465*sqrt(2)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a) )/(sqrt(2)*sqrt(a) + sqrt(I*a*tan(d*x + c) + a)))/sqrt(a) + 4*(3465*(I*a*t an(d*x + c) + a)^5 - 11550*(I*a*tan(d*x + c) + a)^4*a + 7392*(I*a*tan(d*x + c) + a)^3*a^2 + 2112*(I*a*tan(d*x + c) + a)^2*a^3 + 1408*(I*a*tan(d*x + c) + a)*a^4 + 1280*a^5)/((I*a*tan(d*x + c) + a)^(11/2) - 4*(I*a*tan(d*x + c) + a)^(9/2)*a + 4*(I*a*tan(d*x + c) + a)^(7/2)*a^2))/(a*d)
Exception generated. \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeDone
Timed out. \[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:
int(cos(c + d*x)^4/(a + a*tan(c + d*x)*1i)^(3/2),x)
Output:
int(cos(c + d*x)^4/(a + a*tan(c + d*x)*1i)^(3/2), x)
\[ \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx=\frac {\int \frac {\cos \left (d x +c \right )^{4}}{\sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right ) i +\sqrt {\tan \left (d x +c \right ) i +1}}d x}{\sqrt {a}\, a} \] Input:
int(cos(d*x+c)^4/(a+I*a*tan(d*x+c))^(3/2),x)
Output:
int(cos(c + d*x)**4/(sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)*i + sqrt(tan(c + d*x)*i + 1)),x)/(sqrt(a)*a)