Optimal. Leaf size=116 \[ \frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {a \left (a^2-3 b^2\right ) \cos (c+d x)}{d}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a^2 b \cos ^2(c+d x)}{2 d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4397, 2721, 894} \[ -\frac {a \left (a^2-3 b^2\right ) \cos (c+d x)}{d}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a^2 b \cos ^2(c+d x)}{2 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 894
Rule 2721
Rule 4397
Rubi steps
\begin {align*} \int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx &=\int (b+a \cos (c+d x))^3 \tan ^3(c+d x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(b+x)^3 \left (a^2-x^2\right )}{x^3} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (a^2 \left (1-\frac {3 b^2}{a^2}\right )+\frac {a^2 b^3}{x^3}+\frac {3 a^2 b^2}{x^2}+\frac {3 a^2 b-b^3}{x}-3 b x-x^2\right ) \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac {a \left (a^2-3 b^2\right ) \cos (c+d x)}{d}+\frac {3 a^2 b \cos ^2(c+d x)}{2 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {b \left (3 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.29, size = 102, normalized size = 0.88 \[ \frac {a^3 \cos (3 (c+d x))-9 a \left (a^2-4 b^2\right ) \cos (c+d x)+9 a^2 b \cos (2 (c+d x))-36 a^2 b \log (\cos (c+d x))+36 a b^2 \sec (c+d x)+6 b^3 \sec ^2(c+d x)+12 b^3 \log (\cos (c+d x))}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 123, normalized size = 1.06 \[ \frac {4 \, a^{3} \cos \left (d x + c\right )^{5} + 18 \, a^{2} b \cos \left (d x + c\right )^{4} - 9 \, a^{2} b \cos \left (d x + c\right )^{2} + 36 \, a b^{2} \cos \left (d x + c\right ) - 12 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 12 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 6 \, b^{3}}{12 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 164, normalized size = 1.41 \[ -\frac {\cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) a^{3}}{3 d}-\frac {2 a^{3} \cos \left (d x +c \right )}{3 d}-\frac {3 a^{2} b \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {3 a^{2} b \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {3 \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) a \,b^{2}}{d}+\frac {6 a \,b^{2} \cos \left (d x +c \right )}{d}+\frac {b^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b^{3} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 113, normalized size = 0.97 \[ \frac {{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{3}}{3 \, d} - \frac {3 \, {\left (\sin \left (d x + c\right )^{2} + \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} a^{2} b}{2 \, d} - \frac {b^{3} {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{2 \, d} + \frac {3 \, a b^{2} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.52, size = 219, normalized size = 1.89 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-\frac {4\,a^3}{3}-6\,a^2\,b+12\,a\,b^2+2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (4\,a^3-6\,a^2\,b+12\,a\,b^2-6\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {20\,a^3}{3}+6\,a^2\,b-12\,a\,b^2+6\,b^3\right )+12\,a\,b^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (6\,a^2\,b-2\,b^3\right )-\frac {4\,a^3}{3}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^2\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}-\frac {2\,b^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )-6\,a^2\,b\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________