3.3.53 \(\int (a+b \tan ^2(c+d x)) \, dx\) [253]

Optimal. Leaf size=19 \[ a x-b x+\frac {b \tan (c+d x)}{d} \]

[Out]

a*x-b*x+b*tan(d*x+c)/d

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3554, 8} \begin {gather*} a x+\frac {b \tan (c+d x)}{d}-b x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*Tan[c + d*x]^2,x]

[Out]

a*x - b*x + (b*Tan[c + d*x])/d

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

\begin {align*} \int \left (a+b \tan ^2(c+d x)\right ) \, dx &=a x+b \int \tan ^2(c+d x) \, dx\\ &=a x+\frac {b \tan (c+d x)}{d}-b \int 1 \, dx\\ &=a x-b x+\frac {b \tan (c+d x)}{d}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.47 \begin {gather*} a x-\frac {b \text {ArcTan}(\tan (c+d x))}{d}+\frac {b \tan (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*Tan[c + d*x]^2,x]

[Out]

a*x - (b*ArcTan[Tan[c + d*x]])/d + (b*Tan[c + d*x])/d

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Maple [A]
time = 0.02, size = 29, normalized size = 1.53

method	result	size
norman	\(\left (a -b \right ) x +\frac {b \tan \left (d x +c \right )}{d}\)	\(20\)
derivativedivides	\(\frac {b \tan \left (d x +c \right )+\left (a -b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(27\)
default	\(a x +\frac {b \tan \left (d x +c \right )}{d}-\frac {b \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(29\)
risch	\(a x -b x +\frac {2 i b}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*tan(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

a*x+b*tan(d*x+c)/d-b/d*arctan(tan(d*x+c))

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Maxima [A]
time = 0.50, size = 23, normalized size = 1.21 \begin {gather*} a x - \frac {{\left (d x + c - \tan \left (d x + c\right )\right )} b}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*tan(d*x+c)^2,x, algorithm="maxima")

[Out]

a*x - (d*x + c - tan(d*x + c))*b/d

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Fricas [A]
time = 3.34, size = 21, normalized size = 1.11 \begin {gather*} \frac {{\left (a - b\right )} d x + b \tan \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*tan(d*x+c)^2,x, algorithm="fricas")

[Out]

((a - b)*d*x + b*tan(d*x + c))/d

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Sympy [A]
time = 0.05, size = 20, normalized size = 1.05 \begin {gather*} a x + b \left (\begin {cases} - x + \frac {\tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \tan ^{2}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*tan(d*x+c)**2,x)

[Out]

a*x + b*Piecewise((-x + tan(c + d*x)/d, Ne(d, 0)), (x*tan(c)**2, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (19) = 38\).
time = 0.53, size = 231, normalized size = 12.16 \begin {gather*} a x + \frac {{\left (\pi - 4 \, d x \tan \left (d x\right ) \tan \left (c\right ) - \pi \mathrm {sgn}\left (2 \, \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, \tan \left (d x\right ) \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) - 2 \, \tan \left (c\right )\right ) \tan \left (d x\right ) \tan \left (c\right ) - \pi \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \arctan \left (\frac {\tan \left (d x\right ) \tan \left (c\right ) - 1}{\tan \left (d x\right ) + \tan \left (c\right )}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \arctan \left (\frac {\tan \left (d x\right ) + \tan \left (c\right )}{\tan \left (d x\right ) \tan \left (c\right ) - 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 4 \, d x + \pi \mathrm {sgn}\left (2 \, \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, \tan \left (d x\right ) \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) - 2 \, \tan \left (c\right )\right ) - 2 \, \arctan \left (\frac {\tan \left (d x\right ) \tan \left (c\right ) - 1}{\tan \left (d x\right ) + \tan \left (c\right )}\right ) - 2 \, \arctan \left (\frac {\tan \left (d x\right ) + \tan \left (c\right )}{\tan \left (d x\right ) \tan \left (c\right ) - 1}\right ) - 4 \, \tan \left (d x\right ) - 4 \, \tan \left (c\right )\right )} b}{4 \, {\left (d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*tan(d*x+c)^2,x, algorithm="giac")

[Out]

a*x + 1/4*(pi - 4*d*x*tan(d*x)*tan(c) - pi*sgn(2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(
c))*tan(d*x)*tan(c) - pi*tan(d*x)*tan(c) + 2*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c)))*tan(d*x)*tan(c)
+ 2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)*tan(c) + 4*d*x + pi*sgn(2*tan(d*x)^2*tan(c) +
2*tan(d*x)*tan(c)^2 - 2*tan(d*x) - 2*tan(c)) - 2*arctan((tan(d*x)*tan(c) - 1)/(tan(d*x) + tan(c))) - 2*arctan(
(tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1)) - 4*tan(d*x) - 4*tan(c))*b/(d*tan(d*x)*tan(c) - d)

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Mupad [B]
time = 11.44, size = 21, normalized size = 1.11 \begin {gather*} \frac {b\,\mathrm {tan}\left (c+d\,x\right )+d\,x\,\left (a-b\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*tan(c + d*x)^2,x)

[Out]

(b*tan(c + d*x) + d*x*(a - b))/d

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