Optimal. Leaf size=202 \[ -\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x \left (16 b^2 n^2+1\right )}-\frac {4 b n \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (16 b^2 n^2+1\right )}-\frac {12 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )}-\frac {24 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )}-\frac {24 b^4 n^4}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4487, 30} \[ -\frac {12 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )}-\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x \left (16 b^2 n^2+1\right )}-\frac {4 b n \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (16 b^2 n^2+1\right )}-\frac {24 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )}-\frac {24 b^4 n^4}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 4487
Rubi steps
\begin {align*} \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\frac {4 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+16 b^2 n^2\right ) x}-\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{\left (1+16 b^2 n^2\right ) x}+\frac {\left (12 b^2 n^2\right ) \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx}{1+16 b^2 n^2}\\ &=-\frac {24 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac {12 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac {4 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+16 b^2 n^2\right ) x}-\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{\left (1+16 b^2 n^2\right ) x}+\frac {\left (24 b^4 n^4\right ) \int \frac {1}{x^2} \, dx}{1+20 b^2 n^2+64 b^4 n^4}\\ &=-\frac {24 b^4 n^4}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac {24 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac {12 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac {4 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+16 b^2 n^2\right ) x}-\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{\left (1+16 b^2 n^2\right ) x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.51, size = 170, normalized size = 0.84 \[ -\frac {128 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-16 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-4 \left (16 b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (4 b^2 n^2+1\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+8 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-4 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+192 b^4 n^4+60 b^2 n^2+3}{8 x \left (64 b^4 n^4+20 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 162, normalized size = 0.80 \[ -\frac {24 \, b^{4} n^{4} + {\left (4 \, b^{2} n^{2} + 1\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 16 \, b^{2} n^{2} - 2 \, {\left (10 \, b^{2} n^{2} + 1\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 4 \, {\left ({\left (4 \, b^{3} n^{3} + b n\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - {\left (10 \, b^{3} n^{3} + b n\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 1}{{\left (64 \, b^{4} n^{4} + 20 \, b^{2} n^{2} + 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.41, size = 1085, normalized size = 5.37 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^4}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________