3.1.61 \(\int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx\) [61]

Optimal. Leaf size=55 \[ -\frac {2 i (a+i a \tan (c+d x))^7}{7 a^2 d}+\frac {i (a+i a \tan (c+d x))^8}{8 a^3 d} \]

[Out]

-2/7*I*(a+I*a*tan(d*x+c))^7/a^2/d+1/8*I*(a+I*a*tan(d*x+c))^8/a^3/d

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Rubi [A]
time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \begin {gather*} \frac {i (a+i a \tan (c+d x))^8}{8 a^3 d}-\frac {2 i (a+i a \tan (c+d x))^7}{7 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(((-2*I)/7)*(a + I*a*Tan[c + d*x])^7)/(a^2*d) + ((I/8)*(a + I*a*Tan[c + d*x])^8)/(a^3*d)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 3568

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rubi steps

\begin {align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {i \text {Subst}\left (\int (a-x) (a+x)^6 \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \text {Subst}\left (\int \left (2 a (a+x)^6-(a+x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {2 i (a+i a \tan (c+d x))^7}{7 a^2 d}+\frac {i (a+i a \tan (c+d x))^8}{8 a^3 d}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(55)=110\).
time = 0.95, size = 143, normalized size = 2.60 \begin {gather*} \frac {a^5 \sec (c) \sec ^8(c+d x) (35 i \cos (c)+28 i \cos (c+2 d x)+28 i \cos (3 c+2 d x)+14 i \cos (3 c+4 d x)+14 i \cos (5 c+4 d x)-35 \sin (c)+28 \sin (c+2 d x)-28 \sin (3 c+2 d x)+14 \sin (3 c+4 d x)-14 \sin (5 c+4 d x)+8 \sin (5 c+6 d x)+\sin (7 c+8 d x))}{56 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^5,x]

[Out]

(a^5*Sec[c]*Sec[c + d*x]^8*((35*I)*Cos[c] + (28*I)*Cos[c + 2*d*x] + (28*I)*Cos[3*c + 2*d*x] + (14*I)*Cos[3*c +
4*d*x] + (14*I)*Cos[5*c + 4*d*x] - 35*Sin[c] + 28*Sin[c + 2*d*x] - 28*Sin[3*c + 2*d*x] + 14*Sin[3*c + 4*d*x]
- 14*Sin[5*c + 4*d*x] + 8*Sin[5*c + 6*d*x] + Sin[7*c + 8*d*x]))/(56*d)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (47 ) = 94\).
time = 0.25, size = 213, normalized size = 3.87

method	result	size
risch	\(\frac {32 i a^{5} \left (28 \,{\mathrm e}^{12 i \left (d x +c \right )}+56 \,{\mathrm e}^{10 i \left (d x +c \right )}+70 \,{\mathrm e}^{8 i \left (d x +c \right )}+56 \,{\mathrm e}^{6 i \left (d x +c \right )}+28 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{7 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}\)	\(91\)
derivativedivides	\(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{4 \cos \left (d x +c \right )^{4}}-a^{5} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(213\)
default	\(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{4 \cos \left (d x +c \right )^{4}}-a^{5} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(213\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

1/d*(I*a^5*(1/8*sin(d*x+c)^6/cos(d*x+c)^8+1/24*sin(d*x+c)^6/cos(d*x+c)^6)+5*a^5*(1/7*sin(d*x+c)^5/cos(d*x+c)^7
+2/35*sin(d*x+c)^5/cos(d*x+c)^5)-10*I*a^5*(1/6*sin(d*x+c)^4/cos(d*x+c)^6+1/12*sin(d*x+c)^4/cos(d*x+c)^4)-10*a^
5*(1/5*sin(d*x+c)^3/cos(d*x+c)^5+2/15*sin(d*x+c)^3/cos(d*x+c)^3)+5/4*I*a^5/cos(d*x+c)^4-a^5*(-2/3-1/3*sec(d*x+
c)^2)*tan(d*x+c))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (43) = 86\).
time = 0.28, size = 108, normalized size = 1.96 \begin {gather*} -\frac {-7 i \, a^{5} \tan \left (d x + c\right )^{8} - 40 \, a^{5} \tan \left (d x + c\right )^{7} + 84 i \, a^{5} \tan \left (d x + c\right )^{6} + 56 \, a^{5} \tan \left (d x + c\right )^{5} + 70 i \, a^{5} \tan \left (d x + c\right )^{4} + 168 \, a^{5} \tan \left (d x + c\right )^{3} - 140 i \, a^{5} \tan \left (d x + c\right )^{2} - 56 \, a^{5} \tan \left (d x + c\right )}{56 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/56*(-7*I*a^5*tan(d*x + c)^8 - 40*a^5*tan(d*x + c)^7 + 84*I*a^5*tan(d*x + c)^6 + 56*a^5*tan(d*x + c)^5 + 70*
I*a^5*tan(d*x + c)^4 + 168*a^5*tan(d*x + c)^3 - 140*I*a^5*tan(d*x + c)^2 - 56*a^5*tan(d*x + c))/d

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (43) = 86\).
time = 0.38, size = 191, normalized size = 3.47 \begin {gather*} -\frac {32 \, {\left (-28 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} - 56 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 70 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 56 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 28 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 8 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{5}\right )}}{7 \, {\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^5,x, algorithm="fricas")

[Out]

-32/7*(-28*I*a^5*e^(12*I*d*x + 12*I*c) - 56*I*a^5*e^(10*I*d*x + 10*I*c) - 70*I*a^5*e^(8*I*d*x + 8*I*c) - 56*I*
a^5*e^(6*I*d*x + 6*I*c) - 28*I*a^5*e^(4*I*d*x + 4*I*c) - 8*I*a^5*e^(2*I*d*x + 2*I*c) - I*a^5)/(d*e^(16*I*d*x +
16*I*c) + 8*d*e^(14*I*d*x + 14*I*c) + 28*d*e^(12*I*d*x + 12*I*c) + 56*d*e^(10*I*d*x + 10*I*c) + 70*d*e^(8*I*d
*x + 8*I*c) + 56*d*e^(6*I*d*x + 6*I*c) + 28*d*e^(4*I*d*x + 4*I*c) + 8*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} i a^{5} \left (\int \left (- i \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int 5 \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 10 \tan ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int \tan ^{5}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 10 i \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 5 i \tan ^{4}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4*(a+I*a*tan(d*x+c))**5,x)

[Out]

I*a**5*(Integral(-I*sec(c + d*x)**4, x) + Integral(5*tan(c + d*x)*sec(c + d*x)**4, x) + Integral(-10*tan(c + d
*x)**3*sec(c + d*x)**4, x) + Integral(tan(c + d*x)**5*sec(c + d*x)**4, x) + Integral(10*I*tan(c + d*x)**2*sec(
c + d*x)**4, x) + Integral(-5*I*tan(c + d*x)**4*sec(c + d*x)**4, x))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (43) = 86\).
time = 0.87, size = 108, normalized size = 1.96 \begin {gather*} -\frac {-7 i \, a^{5} \tan \left (d x + c\right )^{8} - 40 \, a^{5} \tan \left (d x + c\right )^{7} + 84 i \, a^{5} \tan \left (d x + c\right )^{6} + 56 \, a^{5} \tan \left (d x + c\right )^{5} + 70 i \, a^{5} \tan \left (d x + c\right )^{4} + 168 \, a^{5} \tan \left (d x + c\right )^{3} - 140 i \, a^{5} \tan \left (d x + c\right )^{2} - 56 \, a^{5} \tan \left (d x + c\right )}{56 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^5,x, algorithm="giac")

[Out]

-1/56*(-7*I*a^5*tan(d*x + c)^8 - 40*a^5*tan(d*x + c)^7 + 84*I*a^5*tan(d*x + c)^6 + 56*a^5*tan(d*x + c)^5 + 70*
I*a^5*tan(d*x + c)^4 + 168*a^5*tan(d*x + c)^3 - 140*I*a^5*tan(d*x + c)^2 - 56*a^5*tan(d*x + c))/d

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Mupad [B]
time = 3.30, size = 151, normalized size = 2.75 \begin {gather*} \frac {a^5\,\sin \left (c+d\,x\right )\,\left (56\,{\cos \left (c+d\,x\right )}^7+{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )\,140{}\mathrm {i}-168\,{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^2-{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3\,70{}\mathrm {i}-56\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^4-{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^5\,84{}\mathrm {i}+40\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^6+{\sin \left (c+d\,x\right )}^7\,7{}\mathrm {i}\right )}{56\,d\,{\cos \left (c+d\,x\right )}^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(c + d*x)*1i)^5/cos(c + d*x)^4,x)

[Out]

(a^5*sin(c + d*x)*(40*cos(c + d*x)*sin(c + d*x)^6 + cos(c + d*x)^6*sin(c + d*x)*140i + 56*cos(c + d*x)^7 + sin
(c + d*x)^7*7i - cos(c + d*x)^2*sin(c + d*x)^5*84i - 56*cos(c + d*x)^3*sin(c + d*x)^4 - cos(c + d*x)^4*sin(c +
d*x)^3*70i - 168*cos(c + d*x)^5*sin(c + d*x)^2))/(56*d*cos(c + d*x)^8)

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