Optimal. Leaf size=220 \[ a^3 x+\frac {15}{2} a b^2 x-\frac {3 a^2 b \cos (c+d x)}{d}-\frac {3 b^3 \cos (c+d x)}{d}+\frac {b^3 \cos ^3(c+d x)}{3 d}-\frac {6 a^2 b \sec (c+d x)}{d}-\frac {3 b^3 \sec (c+d x)}{d}+\frac {a^2 b \sec ^3(c+d x)}{d}+\frac {b^3 \sec ^3(c+d x)}{3 d}-\frac {a^3 \tan (c+d x)}{d}-\frac {15 a b^2 \tan (c+d x)}{2 d}+\frac {a^3 \tan ^3(c+d x)}{3 d}+\frac {5 a b^2 \tan ^3(c+d x)}{2 d}-\frac {3 a b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2801, 3554, 8,
2670, 276, 2671, 294, 308, 209} \begin {gather*} \frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {a^3 \tan (c+d x)}{d}+a^3 x-\frac {3 a^2 b \cos (c+d x)}{d}+\frac {a^2 b \sec ^3(c+d x)}{d}-\frac {6 a^2 b \sec (c+d x)}{d}+\frac {5 a b^2 \tan ^3(c+d x)}{2 d}-\frac {15 a b^2 \tan (c+d x)}{2 d}-\frac {3 a b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac {15}{2} a b^2 x+\frac {b^3 \cos ^3(c+d x)}{3 d}-\frac {3 b^3 \cos (c+d x)}{d}+\frac {b^3 \sec ^3(c+d x)}{3 d}-\frac {3 b^3 \sec (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 209
Rule 276
Rule 294
Rule 308
Rule 2670
Rule 2671
Rule 2801
Rule 3554
Rubi steps
\begin {align*} \int (a+b \sin (c+d x))^3 \tan ^4(c+d x) \, dx &=\int \left (a^3 \tan ^4(c+d x)+3 a^2 b \sin (c+d x) \tan ^4(c+d x)+3 a b^2 \sin ^2(c+d x) \tan ^4(c+d x)+b^3 \sin ^3(c+d x) \tan ^4(c+d x)\right ) \, dx\\ &=a^3 \int \tan ^4(c+d x) \, dx+\left (3 a^2 b\right ) \int \sin (c+d x) \tan ^4(c+d x) \, dx+\left (3 a b^2\right ) \int \sin ^2(c+d x) \tan ^4(c+d x) \, dx+b^3 \int \sin ^3(c+d x) \tan ^4(c+d x) \, dx\\ &=\frac {a^3 \tan ^3(c+d x)}{3 d}-a^3 \int \tan ^2(c+d x) \, dx-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^4} \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{d}-\frac {b^3 \text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^3 \tan (c+d x)}{d}+\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {3 a b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+a^3 \int 1 \, dx-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \left (1+\frac {1}{x^4}-\frac {2}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (15 a b^2\right ) \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {b^3 \text {Subst}\left (\int \left (3+\frac {1}{x^4}-\frac {3}{x^2}-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=a^3 x-\frac {3 a^2 b \cos (c+d x)}{d}-\frac {3 b^3 \cos (c+d x)}{d}+\frac {b^3 \cos ^3(c+d x)}{3 d}-\frac {6 a^2 b \sec (c+d x)}{d}-\frac {3 b^3 \sec (c+d x)}{d}+\frac {a^2 b \sec ^3(c+d x)}{d}+\frac {b^3 \sec ^3(c+d x)}{3 d}-\frac {a^3 \tan (c+d x)}{d}+\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {3 a b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac {\left (15 a b^2\right ) \text {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=a^3 x-\frac {3 a^2 b \cos (c+d x)}{d}-\frac {3 b^3 \cos (c+d x)}{d}+\frac {b^3 \cos ^3(c+d x)}{3 d}-\frac {6 a^2 b \sec (c+d x)}{d}-\frac {3 b^3 \sec (c+d x)}{d}+\frac {a^2 b \sec ^3(c+d x)}{d}+\frac {b^3 \sec ^3(c+d x)}{3 d}-\frac {a^3 \tan (c+d x)}{d}-\frac {15 a b^2 \tan (c+d x)}{2 d}+\frac {a^3 \tan ^3(c+d x)}{3 d}+\frac {5 a b^2 \tan ^3(c+d x)}{2 d}-\frac {3 a b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}+\frac {\left (15 a b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=a^3 x+\frac {15}{2} a b^2 x-\frac {3 a^2 b \cos (c+d x)}{d}-\frac {3 b^3 \cos (c+d x)}{d}+\frac {b^3 \cos ^3(c+d x)}{3 d}-\frac {6 a^2 b \sec (c+d x)}{d}-\frac {3 b^3 \sec (c+d x)}{d}+\frac {a^2 b \sec ^3(c+d x)}{d}+\frac {b^3 \sec ^3(c+d x)}{3 d}-\frac {a^3 \tan (c+d x)}{d}-\frac {15 a b^2 \tan (c+d x)}{2 d}+\frac {a^3 \tan ^3(c+d x)}{3 d}+\frac {5 a b^2 \tan ^3(c+d x)}{2 d}-\frac {3 a b^2 \sin ^2(c+d x) \tan ^3(c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.46, size = 226, normalized size = 1.03 \begin {gather*} \frac {\sec ^3(c+d x) \left (-300 a^2 b-210 b^3+36 a \left (2 a^2+15 b^2\right ) (c+d x) \cos (c+d x)-3 \left (144 a^2 b+91 b^3\right ) \cos (2 (c+d x))+24 a^3 c \cos (3 (c+d x))+180 a b^2 c \cos (3 (c+d x))+24 a^3 d x \cos (3 (c+d x))+180 a b^2 d x \cos (3 (c+d x))-36 a^2 b \cos (4 (c+d x))-30 b^3 \cos (4 (c+d x))+b^3 \cos (6 (c+d x))-90 a b^2 \sin (c+d x)-32 a^3 \sin (3 (c+d x))-195 a b^2 \sin (3 (c+d x))-9 a b^2 \sin (5 (c+d x))\right )}{96 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.24, size = 268, normalized size = 1.22
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+3 a^{2} b \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+b^{3} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{3}\right )}{d}\) | \(268\) |
default | \(\frac {a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+3 a^{2} b \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+b^{3} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{3}\right )}{d}\) | \(268\) |
risch | \(a^{3} x +\frac {15 a \,b^{2} x}{2}+\frac {b^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {3 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {3 b \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{2 d}-\frac {11 b^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {3 b \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 d}-\frac {11 b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 i a \,b^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {b^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}-\frac {2 \left (6 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+27 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+18 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+9 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+6 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+36 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+24 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+14 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+4 i a^{3}+21 i a \,b^{2}+18 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+9 b^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{3}}\) | \(337\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.64, size = 167, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} + 3 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a b^{2} + 2 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} b^{3} - 6 \, a^{2} b {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 157, normalized size = 0.71 \begin {gather*} \frac {2 \, b^{3} \cos \left (d x + c\right )^{6} + 3 \, {\left (2 \, a^{3} + 15 \, a b^{2}\right )} d x \cos \left (d x + c\right )^{3} - 18 \, {\left (a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} + 6 \, a^{2} b + 2 \, b^{3} - 18 \, {\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} - {\left (9 \, a b^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{3} - 6 \, a b^{2} + 2 \, {\left (4 \, a^{3} + 21 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sin {\left (c + d x \right )}\right )^{3} \tan ^{4}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 9.20, size = 297, normalized size = 1.35 \begin {gather*} \frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+15\,b^2\right )}{2\,a^3+15\,a\,b^2}\right )\,\left (2\,a^2+15\,b^2\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {2\,a^3}{3}+5\,a\,b^2\right )-16\,a^2\,b-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^3+15\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {2\,a^3}{3}+5\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (2\,a^3+15\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (12\,a^3+42\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (12\,a^3+42\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (48\,a^2\,b+32\,b^3\right )-\frac {32\,b^3}{3}+32\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________